Brand New Fear mode:
Beat the game and save after the credits.

Chainsaw:
Begin a game in Brand New Fear mode. When you enter the forest area for the first time search around a cut tree trunk near Jasper's car.

Submachine gun:
Beat the game and get nine big stars, then begin a new game in Brand New Fear mode. The submachine gun is in the Apartment world in room 202.

Alternate Eileen costume:
Begin a game in Brand New Fear mode. When you get to "Outside of Room 302" enter room 303 and find the costume. Beat the game, then start a new game on the same save for the nurse costume option.

One weapon mode:
Beat the game on hard and get 10 stars.

All weapons mode:
Beat one weapon mode.

Alternate Cynthia costume:
Unlock Eileen's costume and all four endings on one saved game. Then begin a new game on the same save and pick the nurse costume for Eileen.

Thanks to Revolution reader Scary Mike!