This is easiest on level 1-1.
It's entirely possible at the very start of the game, but becomes easier once you've unlocked the rabbit mask (which grants significantly increased movement speed).
Your combo does not diminish in-between arenas, but only whilst in a room with enemies still present.
You'll want to move between enemies as fast as possible, killing as quickly and fluently as possible. Variety does not matter, but obviously some weapons will be easier to use.
If you die, you should respawn at the start of your current area with your combo as it was when you entered the area. So you don't need a flawless run.