Best method for grease and grime under car?

For small parts you can invest in a parts washer. If you have a machine shop near by, you can see if they would hot tank your parts for a small fee. I had my K-member and lower control arms hot tanked. They came out very clean.

As for the body itself, a heated pressure washer will do the job easily. A unit that gets the water very hot isn't cheap. See if you can rent one for the day. A heated pressure washer can even remove undercoating. Some even prefer this method of undercoating removal over media blasting.