Increasing the level cap is one of the core scoring aspects of this game, and you'll definitely need it for later achievements.
As you play notes for a particular instrument, your multiplier for that instrument will go up. It can go up to 3 higher than your current multiplier. Whenever you hit a checkpoint, your level cap will move up so that it is 3 higher than your lowest current multiplier. In order to make it move up by 3x, you have to have every instrument at your current max multiplier.
The best way to go about this is to focus on first maxing out the multiplier for the more sporadic parts. Usually this is the Vocals or the Keyboard (in a guitar heavy song). For most songs, you're going to generally focus on playing Guitar, so I would say to save this for last most of the time.
If you're struggling with this achievement, I'd first make sure to play a song with no Keyboard. Just focus on the parts that have the least notes, and always switch to a different instrument if you have a big pause. There are VERY FEW checkpoints in the game where you can't go up by 3. If a checkpoint contains none or very little of a particular instrument, it will be greyed out, and doesn't count towards raising your multiplier.