You might think that there's no strategy to come up with in a game that only has 2 buttons for controls, but this is not entirely correct. With the vast amounts of enemies and the high speed Grandmaster forces on you, it is mandatory to understand certain game mechanics better and use different skill builds than you probably did on student and master.

First and foremost, you should know about time deceleration. Time is slowed down when certain conditions are met, most notably when enemies are in both of your attack zones. Other than that, time will also slow down whenever you hit, are hit or perform certain finishing moves (that you cannot influence). These moments are a great opportunity to get a quick overview and determine your next steps. You have to avoid getting hit as much as possible, having more than 5 health left after killing half of the stage is nice because stages tend to get a little harder during the last 100 kills or so.
Also, do NOT wait for a colored enemy to pop up when you're holding a ranged weapon and the stage is already at high speed, or a grey enemy is quite likely to give you some spanking. It's generally better to throw one in each direction when you pick up more than one ranged weapon at once, too.

For skills, I would recommend 3 that don't really make a lot of sense on lower difficulties: Heal, Deflect Grey and Deflect Color. They increase your effective health, and just like Survival I usually combine them with Grey Out and Light 'Em Up.
Other skills also work decently such as Iron Fist, Venom Fist, Master Brawler and maybe Unarmed Master too, but just like the weapon skills I think it's better on master or student, partially because it tends to proc at the same time or shortly before Light 'Em Up, much like Freezing Point and Deep Impact, both of which seem too random to be as useful as the ones I recommended. I can see Throw Weapon working nicely as well because of the low kill requirement.

With this build, try to kill as many enemies as possible without getting hit (well, duh) because you can charge up all your skills faster which is sure to buff your survivability. Go wild whenever Grey Out and Light 'Em Up trigger roughly at the same time!

I'll just focus on how to progress the map as quickly as possible here because farming medals and perfect rounds is much more reasonable on student and master difficulty anyway. While this is somewhat preferential, there are a few routes that require you to play more levels than others, not to mention that certain routes only exist so you can unlock all the skills (which carry over between difficulties). Also, I personally prefer other rounds over Mob Rounds simply because they are faster to clear, but this is already preferential and you can choose any pathing you like as long as it gets you to the final Boss Round.

Bomb Rounds are fairly straightforward. You only have 1 HP, your skills don't work and you are restricted to throwing, well, bombs. The last few can be quite hard due to their high speed, so you'll want to lean back, get a good view of the entire screen and react to the amount of enemies poppin up on either side. Never waste a bomb throw, or you'll most likely die.

Dagger Rounds work exactly like Bomb Rounds, just that you're throwing Daggers (duh).

Boss Rounds require you to fight 1 or 2 enemies that change their color (and thus their attack patterns) every time you beat them until their health bar is emptied. This requires a bit of focus but should prove easier than most other rounds. To state the obvious, you cannot use skills.

Defender Rounds are quite similar to Bomb & Dagger Rounds (1 HP, no skills, only one way of attacking), but instead of throwing things you are supposed to deflect daggers thrown at you by grey dudes. For this reason you are equipped with a sword at all times, and as long as you don't miss (!!!) you'll be fine. Just abuse the hell out of the time deceleration effects to plan ahead carefully.

These work like simplified but incredibly fast Mob Rounds. Your range is increased because you're holding onto a melee weapon at all times, and all enemies but the last one (a Brawler) will die in one hit. The only disadvantage is that these rounds start at high speeds and actually increase their speed the more enemies you kill. Getting to the Brawler on full health will prove almost impossible, but you should make sure you have at least 5 HP or so remaining when you get there - I have yet to see someone who can perfect the boss in the last few light sword and nunchaku rounds (lava area) on Grandmaster. Also, skills obviously don't work here.
To cope with the challenging pace of these rounds, try to focus even more on your range indicators than you usually would (but don't go tunnel vision mode!). Missing usually means you'll get hit which might anger you and disrupt your focus even more.

By far the most common kind of round, but far from the easiest. On Grandmaster, enemies are not just fast. Their numbers will make this more of a battle of endurance than anything else - if you lose focus after killing half of them you can still die before reaching the end.
Once you're about halfway through the map, Mob Rounds will start having 300+ enemies each, a few actually skyrocket to 420 or even 500 (highest I've seen so far). The final one you have to beat in order to progress through the lava area has 360, and the only one appearing after it has 425.

On Grandmaster, Multi Rounds will have 20 mobs per round, and 6 rounds per level. These are not too hard, but seeing as each consecutive round will be faster you should try to memorize which enemies come when and from which side. Finding the right strategy (and executing it faster each round) is the key to beating these, which shouldn't be too hard in general.
Also, please keep in mind that while skills do work here, your skills have their kill counter reset for each round. This means that all skills that do not activate before you hit 20 kills will not trigger at all. Personally I beat these without any skills because I was too lazy to change them, but if you want to you can go with Three Arrows/Daggers/Bombs and Brawl Master because all of these only require one kill before you can use them (still have to get the respective weapon though).
In the video I actually did change my skills to verify their usefulness.

These are fairly easy despite the fact that skills don't work on them. All you really have to do is keep fighting until the required amount of objects has been smashed (the last Smash Rounds in Grandmaster requires 40 smashed objects). You can probably beat most of these on your first try.

On Grandmaster, Speed Rounds will always give you 50 seconds to deal with 90 enemies. While they are relatively short and therefore not too hard, they offer a very low tolerance for failure. Taking a single hit can already mean failure, and missing also slows you down (and is likely to cause you getting damaged). However, 90 enemies is not much and using ranged weapons on multihit enemies and Brawlers saves a lot of time. Just keep trying and don't forget that your skills won't work here.

These always count as a win, even if you finish with 0 kills. You can even skip survival rounds entirely because clicking on them will unlock the level right next to them.

In these rounds you cannot see the color of your enemies, but you will see the hit indicators below them just as usual. Thunderstorm rounds also make it harder to see weapons, but on the bright side you are guaranteed to not run into any Brawlers here. Retro Film Rounds just make your whole screen appear grey but work like ordinary Mob Rounds otherwise. Grey Out (ideally combined with Light 'Em Up) is even more useful here than it is in normal Mob Rounds.
The last Thunderstorm round features 325 enemies while the last Retro Film Round has 380.

I wrote everything here myself, so if I was to credit anyone else it would be the developers for making this awesome game and providing useful loading screen messages. :p

I've already completed level mode on all difficulties and obtained most other achievements, so all that's left for me is survival, and grinding mastery in survival.

If you want (more) gameplay videos for certain rounds, let me know in the comments. I'm glad to hear feedback and suggestions any time as long you know how to say it politely. :)

Also, please consider rating up and faving my guide. If you want to thank me, you could also do the same for my videos here: http://gtm.steamproxy.vip/id/takku/videos/?appid=264200

Software used for the videos:
Fraps (Recording)
VirtualDub (Joining)
Handbrake (Encoding)
Google Chrome (Uploading)