This can be done easily on Normal difficulty, though you will probably want to play through the game once on Hard to not only get a feel for the game, but to get all the collectibles during that run.
For Normal, you can speed past the majority of the enemies in the game. You will only have to fight bosses and rooms that lock until you kill everyone in the area. Abusing Option-A (L1 + Triangle) will only make this easier for you. Any time spent paused or in menus will not count towards your time, and if you die your counter will reset back to the time you were at the nearest checkpoint.
As for upgrades, I would only go for ones that aren't too far off the main path as you are only wasting time, and only need very few of them to make it through the game.