+Failing a counter will cause an enemy nearby to hit you, this is useful if you're playing in single player and want to reset to the last checkpoint quickly, or someone needs revives.

+Higher difficulties on single player will spawn more grimm, which highly speeds up all the challenges that requires killing a number of a type of grimm or android.

+Ursas (both Majors and Minors) will stun themselves after their ranged rock attack.

+Hitting a stunned enemy with a fully charged Heavy attack will always trigger a team attack (except on Ursa Majors and Mutant Beowolves).

+An enemy that has the team attack symbol above their head takes bonus damage as if they were stunned, speeds up the killing of Ursa and Alpha Creeps/Beowolves, as counter->stunned->lots of attacks.

+Stunned and Frozen take damage as if their guard was down, which only effects melee combos.

+For the "kill # of enemy" requirements, you do need to land the killing blow, assists do not count as kills.

+Turret kills do count as kills towards progression. (Credit to tak4n, Roman Tyrianweed).

+Know your combos, and approximately how much damage each does. Having a dodge at the start of the combo can affect it, eg Weiss's Dodge->LH can 1 cycle Creeps if she lands all 3 blows.

+My strategy for dealing with Mutant Beowolves solo is to stand right in their face, counter them when they try to swipe you, then hit them with a fully charged heavy while they're stunned, and repeat. It's a very small window for the counter but once you get the timing down this is very easy and reasonably fast.



+Check the comments for people probably pointing out things I've missed.

Thanks to RoosterTeeth for their rank requirements guide in the discussion section.


Patch 1 notes:
+First encounter on chapter 1 on Eclipse difficulty is your friend for hitstreak challenges
+Brief testing seems to add approximately an extra multiplyer to the number of Grimm.
ie Normal: 1x eg 9
Hard: 2x eg 18
Legendary: 3x eg 26
Eclipse: 4x eg 35
However does not apply as nicely to more "harder" monsters, eg I could only spawn up to 2 Alpha Beowolves.

Requirements:

• Reach Max Level in 4 characters.

Reach level 10 in four characters, simply play until you've managed it. If you've played pre-release you might have some characters already counted as max level.

Requirements:

• Leveled Up - Get 1 character to level 10
  
• 10x Revives - Revive 10 teammates

Needs to be done on multiplayer. The later 5 will be faster thanks to unlocking the Medic skill. Yay acheivement if you haven't already gotten it.

• 50x Ranged Kills - Get 50 kills with ranged attacks

Weiss's Tier 2 ranged is great here, it can 1-cycle creeps, easy on the first level with a higher difficulty on single player.

• 50x Hit Streak - Get a hit streak of 50 or higher

Blake's Stun Blades and her level 10 skill "Fortunate Faunus" are useful for getting this.

• Ursa Slayer - Kill 20 Ursas

Only 20, but if you do still need it after all the above, chapter Five has quite a few spawns for them, especially if you take the right path, or Horde mode.

Requirements:

• Leveled Up - Get 1 character to level 10

• Creep Killer - Kill 300 Creeps

Chapter 1 on Eclipse difficulty (and Ruby with her Ultimate and her LLLH) if necessary.

• 20x Team Attack Finishers - Perform 20 Team Attacks

Needs to be done on multiplayer to count, but won't take you very long.

• 50x Counters - Perform 50 Counters

Easy on single player, get 4 Beowolves (eg first encounter of chapter 6) and you can counter them in a circle.

• 4x Challenge Master - Unlock 4 Mastery Challenges

Mastery Challenges are the challenges you do to unlock extra skills. You'll unlock one for hitting level 10 in each character, but Hunter (hitting level 5) does not count towards this (thanks to Masquerade for the confirmation).
My preferred ones are the counter ones (using the above exploit with the Beowolves), Slugger (Heavy attack kills), Barrage Mastery with Weiss, and the one for the 3rd tier of team attack.
+Simple (but not necessarily fast) Mastery Challenges to complete in single player are:
-Slugger: Simply only use heavy attacks, or use a combo that 1 cycles Creeps and ends in a heavy until enough kills. (Could work well with Jaune's/Ren's Heavy Attack specialty unlocks.)
-Counter Masteries: Get 4 Beowolves (eg first encounter of Chapter 6) then counter them repeatedly until they're dead, takes a handful of sets of Beowolves to get the required number.
-Team attack level 3: Playing in single player you set up team attacks for yourself- with Slugger unlocking the upgrade that improves your chances of triggering team attacks, you'll find it's very common- simply learn what combos of damage you need to do so your team attack deals the final blow.
-Ranged attack level 3: Just only use ranged attacks, or nearly kill enemies then shoot them.
-If you have Yang at level 10, her Second Wind can be abused to gain Ultimate kills (for her Ultimate Tier 3 unlock, and the Improved Ultimate 2).
-If you venture into multiplayer Savior (revive 5 times) is quite common to get.

Requirements:
• Leveled Up - Get 1 characters to level 10
  
• Beowolf Buster - Kill 300 Beowolves

Eclipse difficulty and the start of chapter 6, or the first few waves of Emerald Forest on Horde.

• 20x Team Attack Set Ups - Set up 20 Team Attacks

Needs to be done in multiplayer, have another player execute attacks you set up. Ruby and Jaune are good for setting them up. So is Heavy Synergy from the Slugger challenge.

• 100x Destructibles - Break 100 crates or pots

Good in multiplayer or singleplayer.

10x Artifact Collector - Collect 10 artifacts

Requirements:
• Leveled Up - Get 1 characters to level 10
  
• Alpha Creep Killer - Kill 150 Alpha Creeps

Save this for last, but focus on getting as many Alpha Creep kills as you can while doing the rest.
User Zila points out the waved encounter at the end of chapter 7, as it spawns Alpha Creeps every wave. Make sure to keep at least one enemy alive on the final wave to get KO'd by and reset the encounter to the first wave.
Upping the difficulty on single player will spawn more.

• 5x Flawless Waves - Get 5 Flawless Waves

Can only be done at the end of Chapters 1 and 2 and in Horde mode. Using Blake and her Stun blades simplifies this.
Another good strategy is using a Character who has good aim with their Ranged (Eg Ruby, Pyrrha, Weiss) and using that to aggro Grimm, then destroying them with Ults (Ruby's Cresendo Forte can 1-shot Beowolves) or melee combos once they no longer threaten the Node/Center.

• 20x Revives - Revive 20 teammates

Again, needs to be done on multiplayer.

• 100x Ultimates - Perform 100 Ultimate Attacks

Yang and her Second Wind ability can be abused for this, as it's performing it 100 times, not kills or anything specific.

Lots of Info here.

Requirements:
• 2x Leveled Up - Get 2 characters to level 10
  
• Alpha Beowolf Buster - Kill 150 Alpha Beowolves

Again, keep the Alpha Beowolf kills in mind while completing everything else.
+User Haxipaw suggests Chapter 4, the encounter after the mutant creeps. As Patch 1 increased the checkpoints, you can now just jump straight off the ledge and either Alpha Beowolves or normal Beowolves will be the first spawn. If normal Beowolves spawn, get KO'd, if Alpha Beowolves spawn, kill them then get KO'd to the next grimm. My recommended method.
+User Br0_Inat0r suggets Chapter 3, the waved encounter at the gates. Waves 2, 3, and 4 have a chance of spawning Alpha Beowolves, and the last encounter will always be Ursai to get KO'd to, which will reset the encounter to fight the waves with Alpha Beowolf potential again.
+The way I did this was more grindy. The 3rd encounter of chapter two (where the path splits) is just after a checkpoint. There are 4 things that can happen here from what I've seen (numbers from single player, normal difficulty):
1. Left path: 4 Beowolves and 2 Alpha Creeps
2. Left path: 4 Beowolves, then 4 Beowolves and a slight delay of an Alpha Beowolf
3. Right path: 4-5 Beowolves, then an Alpha Beowolf
4. Right path: 4-5 Beowolves, then 4 Creeps and a slight delay of an Alpha Beowolf
So note the spawn, then either get KO'd to the enemies or kill everything except one- then kill the Alpha Beowolf- then get KO'd to the one you left.
+Rememer you can increase the difficulty if you're confident for bigger spawns. Eclipse gives you 3 Alpha Beowolves for the Chapter 4 or Chapter 3 option, and 2-5 for Chapter 2's option (depending on which version you get).

• 6x Challenge Master - Unlock 6 Mastery Challenges

Your two max level characters will unlock 2 mastery challenges, (but not one for Hunter, level 5), so it's really only an increase of two mastery challenge from rank 2->3 for you to grind out. Check my notes there for detailed advice.

• 10x Wave MVPs - Get 10 Wave MVPs

A Wave MVP is hopefully something you've seen by now, it's when you contribute the most to that wave (reviving is a large contribution). Wave MVP in any mode works.
The Campaign chapters that have waved encounters are:
Chapter 1: 4 waves (last encounter)
Chapter 2: 4 waves (2nd encounter), 4 waves (last encounter)
Chapter 3: 5 waves (2nd encounter)
Chapter 5: 5 waves (last encounter)
Chapter 6: 4 waves (last encounter)
Chapter 7: 4 waves (last encounter)
Chapter 8: 4 waves (last encounter)
Chapter 10: 4 waves (2nd encounter)
If you're focusing on MVP I recommend two things:
1. Run through up to Chapter 3's waved encounter, then restart to Chapter 1. That's 17 waves and you don't have the gap of Chapter 4.
2.a. Play as Ruby and upgrade her ulitmate as much as possible (including extra ultimate pool and hyperballistic) and be RUTHLESS. Her LLLH can 1-shot nearby Creeps and Cresendo Forte can 1-shot Beowolves if they start close enough to her. (Thanks to gvk for pointing out how OP Ruby is)
2.b. Play as Jaune as his teamwork score gets a huge boost from his Ult (also if you're playing with people who like Team Attacks).
Evidence suggests Turret MVP does not count towards this.

If you want more information, here[grimmeclipse.wikia.com] is the wiki's page on how score is calculated- as you can see kills are worth a lot more than assists, so focus on those.

• 15x Artifact Collector - Collect 15 artifacts

This Alpha Beowolf challenge took me the longest, with max level 4 characters a close second, and 150 Alpha Creeps a close third, this is basically the hump in the middle that you need to get over.

Requirements:
• 2x Leveled Up - Get 2 characters to level 10
  
• Boarbatusk Butcher - Kill 100 Boarbatusks

The first encounter of Chapter 2, I recommend single player with the highest difficulty you manage manage them on. If you get sick of Port's cutscene, the 3rd encounter of Chapter 3 has a 50/50 chance of being the righthand path which spawns the same number as Chapter 2.

• 100x Stuns - Stun 100 Grimm

Easy with Blake's Stun Blades, or Ruby's Electric Blast.
You can also stun with countering (Pyrrha's Richochet can buff this), Yang's Concussion Missiles or some of her Ultimates, Weiss's Improved Nova, Nora's electric attacks (eg LLH, don't know if consistent), or Ren's Improved Aura Charge, or Draining Aura Charge.

• 100x Team Attack Finishers - Perform 100 Team Attacks

Need to be done on multiplayer.

• 50x Assists - Perform 50 assists

Needs to be done on multiplayer, attack something that another player gets the killing blow on.

• Shadowverse has pointed out that as Boarbatusks, Ursai and Ursa Majors damage themselves when they become stunned after a missed charge or a rock attack, you can get an assist on single player with patience. To do this, attack one until it has very low health [but above 10] and then wait and dodge their charge or rock attacks until they take themselves out.
  
  

Requirements:
• 2x Leveled Up - Get 2 characters to level 10

• 30x Ursa Slayer - Kill 30 Ursas
  
• Mutant Beowolf Buster - Kill 20 Mutant Beowolves
  
• Android Annihilator - Kill 200 Androids

These 3 synergise well, run through Chapters 8->10 repeatedly, or the later Horde stages.

• 8x Challenge Master - Unlock 8 Mastery Challenges

See rank 2->3 for my advice on mastery challenges. You'll already have accomplished 2 for level 10 for the two characters (but not Hunter for level 5).

Requirements:
• 2x Leveled Up - Get 2 characters to level 10

I recommend Ruby or Jaune be one of these, as it'll make getting MVP much, much easier.

• Mutant Creep Killer - Kill 100 Mutant Creeps

If you do need to grind out the Mutant Creeps, go to a room with Serum tanks to destroy:
1st room: Play as Weiss, destroy the tanks spawning basic Creeps, then stand and shoot the Mutant Creeps, her basic range 1-hits them. (Chapter 9)
2nd room: This room only spawns mutants, play as Blake, her Horizontal Blade will shoot through the tanks and only hit the Mutant Creeps, but also 1-hit them. (Chapter 10) (Thanks Retheron)

• 200x Stuns - Stun 200 Grimm

Blake and her Stun Blades again.

• 75x Hit Streak - Get a 75x hit streak

Blake is good for this, due to Stun Blades and Fortunate Faunus. I found Ren also tended to get high hit streaks, due to many of his attacks dealing multiple blows.

• 50x Wave MVPS - Get wave MVP 50 times

See rank 5->6 for my tips on getting MVP.

The final stretch!

Requirements:
• Maxed Out - Get 4 characters to level 10
  
• Grimm Reaper - Kill 3000 Grimm

Time consuming, but Eclipse and the early levels can speed this up drasticly. I recommend doing this last as everything else contributes to it. Androids do count as Grimm. Turret kills count towards this (Thanks Roman Tyrianweed, tak4n).
Grimm Gauntlet could be fast for this, if you can manage to get a good turret setup that helps you take out waves quickly. Would probably peak around Onslaught 2, especially on the Mountain Glenn map. Reset once the waves become a slog.

• 100x Hit Streak - Get a 100x hit streak

The 100x hit streak- congratulations on the acheivement if you don't already have it- is best done with Blake with Fortunate Faunus- the doubling of time between hits means her combo can continue across waves, and I'd recommend the waved encounter of Chapter 3 for this. Note: Crates/Jars count towards this.

• 10x Challenge Master - Unlock 10 Mastery Challenges

Keep in mind that maxing out 4 characters will net you 4 Mastery Challenges, (not the level 5 challenge), and the 3000 Grimm challenge will get you the kill 2000 Grimm Mastery challenge, so you only need to do 5 optional ones. Depending on your play style you might have the ones for ultimate kills, heavy attack kills, or reviving teammates. If you've done a lot of team attacks you might be close to doing 100 kills with one character's team attack so keep that in mind when choosing what ones you're going to unlock. See rank 2->3 for more detailed tips.

• 50x Flawless Waves - Get 50 Flawless Waves

Like rank 4->5 you can abuse ranged stuns to get flawless waves, or use Horde.
If doing this in Horde solo, Blake and her stun blades on the first few waves of Emerald Forest are lovely.
Deliberately failing a counter to get KO'd on the last wave for the Chapter 1/2 control centers allows you to get 3 waves an attempt without having to re-run the whole level.