Bullet Beat

Start Endless Mode or Endless Hell Mode. Shoot enemies fast enough so the counter continues to increase and does not reset. You may have to wait a level or two before you can reliably keep the counter advancing. However, soon the screen fills with enemies. Your power ups will give you the offense you need to multi-shot them continuously. Your challenge at that point is simply to stay alive until the counter reaches 4000. Expect a lag before the achievement pops.

If you have issues--achievements not popping or the game crashing, take the following remedial action. Uninstall the game. Do a power cycle on the console to clear the cache (Press on the console for about 10 seconds, disconnect the power from the wall socket for about a minute, reconnect power, & restart the console). Reinstall the game. The achievement should pop as described in this guide. Reinstalling the game may be sufficient to remedy the problem. However, just to be on the safe side, you should clear the cache as well.

Edit 6 May 2022: For people who continue to have issues after applying the above remedy, please see the remedy posted by @Vudix at Get Unobtainables Fixed!. You must revoke the game's permissions. When you replay it, the game will again prompt for permission. When you grant it, problematic achievements should pop.

Endless Mode is a little less challenging than Endless Hell Mode. Endless Hell mode, because of its greater challenge, is quite a bit faster. I recommend Endless Hell Mode. The faster you can get the combos the less chance there is of a slip even if the skill level required is a little higher. I tested both ways. On Endless Hell mode, it was pretty quick. It took about 3 minutes. The achievements all unlocked at timestamp 8:29 & 8:30. In contrast, on an alt account, I had to get above level 50 in regular Endless Mode to get a > 7000 combo for all achievements. It took about 10 minutes.

Tip: If you have an arcade stick, use it. The game is far less challenging if playing with an arcade or fighting stick.

Concentrate on grabbing power ups. If your offense is high, enemies will not be able to approach you. Especially, look for shields. As soon as you get a shield, aggressively mop up all other power ups on screen. Ideally, once the screen fills with enemies, you'll get enough shield drops to shield yourself continuously. At that point, the combination of a high multi-shot offense and a shielded ship should allow you to survive indefinitely. Doing so, will earn you the next four achievements in Title Update 4 for Bullet Beat: