Holy hell... Well, to start off with, Dennis can be found at the end of New Gun City. The only relatively safe platforms available for you to shoot him on are green and red conveyor belts which go left and right respectively. There's no place else to stand, so I hope for your sake that you're good at multitasking.
Dennis is an alien who flies a UFO back and forth just above the aforementioned conveyor belts, while firing progressively faster red orbs of doom at you. All in all, they aren't too difficult to dodge as long as you don't take too long getting the last of his health down. I used the 2-way diagonal shot gun and fired upwards while moving against the conveyor belt to keep from falling. Since he was the first boss I killed, I was limited on my choice of guns, so there might be a better method out there.
Do whatever you're comfortable with, but I mostly stayed on the upper-left belt and did what I could to dodge and/or jump over the orbs. As far as the rest of the level leading up to him goes, you pretty much just have to practice and have patience. Three enemies to watch out for in particular:
1. A little yellow-orange vacuum-cleaner dude who launches fireballs at you diagonally. Take him out from far away whenever possible. He's armored well enough that killing him close up between his attacks can be difficult. There's one time you don't have much choice, just after you jump over a small dumpster, but you might get lucky and be able to jump over the fireballs while on top of the dumpster.
2. A yellowish flying cat head... thing. Nothing too terrible, but their fast U-shaped flight pattern can catch you off guard. Keep your distance and they won't fly towards you at all.
3. A large, two-headed, fire-breathing bird. These guys are total bastards. Their flame breath is (sort of) U-shaped also, and the three you encounter throughout the level are all placed in spots you really would rather not have them. Places where if you're not careful about your approach, they'll hit you and most likely cause you to fall and die. First and third are right after a series of conveyor belts, forcing you to deal with them on the belts. The second one is shortly after you go through a white building and the camera shifts full screen. Kind of a mid-level boss, although I suppose you could see them all as quarter level bosses.
Anyway, that's enough out of me. Good luck, and happy platforming!