Power washing would be an acceptable first step but first, use a scraper to get any heavy gunk off.
Any stubborn heavy grease left behind after power washing I would spray with brake cleaner and hit it with a wire wheel on a drill. Prior to painting anything, I like to go over parts with brake cleaner and paper towels to remove any oils that can make the new paint fish eye. For any rusty areas, get a spray can of Etch primer, stuff works great to keep the rust away. Then just top coat with the black of your choice.
In your other post you mentioned using engine enamel but that would not be needed for the compartment. I suggest NOT using gloss black. This would be a personal thing with me but I never cared for engine compartments done that way, they always look too amature. (just a personal thing)
There are a few different choices for paint but to keep it simple, I suggest semi flat Krylon. It's easy to find and at a low cost. It also touches up very well since you'll be driving this car. Other choices could be products from Eastwood, POR, Rustoleum, etc. Everyone has their favorites so you may get other sugggestions there.
Now depending on what your after, I like to use a few different colors to set things off. Parts like the hood hinges, striker plate, master cylinder, steering box and some steering components look good when done in a gray (phosphate look) or cast iron gray.
Here's a before and after shot of the 327 in my Impala. The before shot already had the air cleaner sand blasted, primed and painted so disregard that. (that's the Krylon by the way)
What kind of gloves do y'all use? I can handle bleeding, but it slows me down b/c it makes everything slippery.