I refer to achievements a few times in this guide since I assume that most people won't care to beat Bloody Palace unless they want the achievements for it, considering that it's not exactly fascinating to play this mode for a prolonged amount of time, which is required to beat it no matter how good you are, even though you'll obviously always take the +100 levels portal.
If you don't care about achievements, you can obviously feel free to ignore these bits.

The first game has no Bloody Palace mode, no matter how hard you look for it. In a way, you are already 33% done. Congratulations!

If you care, here's an excellent guide I used while playing for all collectibles and achievements: https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=2934304748

Beat the game on Dante/Lucia Must Die, load that savegame as Trish and start Bloody Palace while holding all shoulder buttons on your gamepad (RB+RT+LB+LT/R1+R2+L1+L2) until you spawned with a full DT gauge bar. Then activate DT and use Air Raid (Jump > Jump > Hold Shoot) to kill everything effortlessly.
There are cheatcodes that let you skip the preparations entirely, but I really recommend to play through the game since you paid for it and you can always use items to heal if necessary.
Also, I don't know if they void any achievements, if that matters to you. It's up to you however, and you can easily find these cheat codes on your favorite search engine or other guides, such as this one I personally read before playing DMC2: https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=2941112591

1. Beat the game with Dante or Lucia on Normal.
   
2. Beat the game with the other character on Normal.
   
3. Optional: Make sure your health bar is maxed out (DT bar doesn't matter but is likely to be maxed at this point) by buying all Blue Orbs, collecting all fragments and completing all Secret Missions**.
   
4. Beat the game with Dante or Lucia on Hard.
   
5. Beat the game with the other character on Hard.
   
6. Beat the game with Dante or Lucia on Dante Must Die or Lucia Must Die, respectively*.
   
7. Optional: Beat the game with the other character on Dante Must Die or Lucia Must Die, respectively*. I personally didn't since it's not required for achievements, and playing DMC2 5 times is more than enough as it is.

* Lucia's campaign is shorter (by 5 missions) and arguably easier, but the underwater boss seems to stay invisible for the entire fight no matter what, so brace yourself.
** For collectibles, easy Red Orbs farming and more, see here: https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=2940707560

1. On the HD Collection's main menu (i.e. not the DMC2 main menu!), set your language to a European one. I recommend German for the meme value.
   
2. Load your latest save game that has Dante/Lucia Must Die mode completed with Trish as your selected character.
   
3. Select Bloody Palace while holding all shoulder buttons on your gamepad (RB+RT+LB+LT/R1+R2+L1+L2) until you spawned with a full DT gauge bar (yes, it's a cheat, but it will not void achievements or anything - feel free to do it legitimately though).
   
4. Equip Aerial Heart. For the other slots I recommend Healing Heart and Frost Heart.
   
5. Activate DT and use Air Raid (hover with Aerial heart while shooting) to melt everything with Trish's death beams of justice.
   
6. Always take the green portal to ascend 100 levels at a time.
   
7. Note that Bloody Palace has no end in this game. Level 9999 will just be repeated indefinitely, so just afk until you die*.
   
8. Get annoyed that your Gold Orb was triggered and afk until you die once more.

* You could just ragequit at this point if you don't care about the Red Orbs bonus at the end.

Super Legendary Dark Knight costume (beat Dante Must Die as Dante) + Quicksilver (beat Mission #12 boss). Enemies don't seem to increase in difficulty, which I believe equals Hard difficulty. From floor 9000 onwards, they spawn with active Devil Trigger (which you will recognize if you played on Dante Must Die or the Secret Mission with the elevator), but it's still next to impossible to die with this setup.
There is a cheatcode that lets you unlock all costumes instantly, but it will void achievements as well as saving and loading, so I didn't even bother trying it. You can search for it if you feel like playing without these features, but I recommend actually playing the game as it's a lot of fun (even more so than DMC1 and definitely more than DMC2) and not too difficult, especially since you can max out your gear on Easy and just keep farming Red Orbs there in order to play every level with maxed out Vital Stars and 3 Gold Orbs (use these for bosses mostly), if it's necessary.

1. Start a playthrough with any character*, preferably in Gold Orb mode (although that's optional).
   
2. Beat the game on Normal**.
   
3. Optional: Make sure your health bar is maxed out**** (DT bar doesn't matter but is likely to be maxed at this point) by buying all Blue Orbs, collecting all fragments and completing all Combat Adjudicators & Secret Missions (all of this can be done on Easy if necessary).
   
4. Beat the game on Hard.
   
5. Optional: Beat the game as Dante on Very Hard to unlock the first Super costume (if you want to make the next step easier).
   
6. Beat the game as Dante on Dante Must Die*** (using Super Dante if you unlocked him in the previous step).
   
7. Make sure your Devil Arms (I used Rebellion and Beowulf) and Guns (I used Ebony & Ivory and Kalina Ann) of choice are maxed, although this should have happened a couple playthroughs ago.

* I recommend Dante so you upgrade him right away, but you can use Vergil for everything up to and including Hard. Note that you must beat Dante Must Die as Dante once for what comes later.
** If Normal is too difficult, you can farm Mission #3 on Easy (unlocked by dying 3 times on Normal) to max your Styles while farming Red Orbs at the same time (break everything in the bar room). You can switch to Gunslinger right before the boss, stand at one of the sides and lock on the heads (switch when they pop) while being semi-afk, maybe dodging the ice rain attack by siderolling. It's better to beat Mission #7 (any difficulty) first to unlock Devil Trigger, if you haven't done so already.
Alternatively, if you only want Red Orbs, Secret Mission #5[devilmaycry.fandom.com] (available during Mission #9) can be repeated indefinitely to get Red Orbs even quicker and easier.
*** Go back to farming Red Orbs if necessary so you can start every level with 3 Gold Orbs and 30 Vital Stars L, just to be safe.
**** For collectibles, see here (NOTE: at the time of writing this, it lists a "Blue Fragment #2" in Mission 4 which does not exist - the mission only has one blue orb fragment, labeled "Blue Fragment #3"): https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=1427559215

1. Use the Super Legendary Dark Knight costume.
   
2. Equip your fully upgraded Devil Arms and Guns of choice.
   
3. Equip Quicksilver for your Style.
   
4. Effortlessly kill everything using your infinite Devil Trigger and infinite Quicksilver.
   
5. Always take the fire portal to ascend 100 levels at a time.
   
6. There's a bossfight on every 10th level you'll play, with the last one being the 3rd Vergil fight, but even these are impossible to fail with permanent DT + Quicksilver.
   
7. Bonus: Clear Heaven or Hell after (or before) finishing Bloody Palace. It's ridiculously fast since everything dies in 1 hit including bosses. You also die in 1 hit, but with Quicksilver and Ebony & Ivory, you'll be fine if you're just a bit careful.
   Especially Mission 18 is great on this difficulty if you still managed to have any Styles that aren't maxed out.
   
8. Bonus: You probably have all achievements now. If you're still missing the one to clear every mission with an S rank*, use Super Dante or Super Legendary Dark Knight on Easy. Or don't, I guess. Vergil also works fine for this, especially with Super costumes.

* This might be helpful (NOT valid for Vergil): https://gtm.steamproxy.vip/sharedfiles/filedetails/?id=1933807836