I would definitely buy the following, in order of importance:
- Sliding dual compound 12" Miter Saw

- 7-1/4 Circular Saw
- 2HP Variable speed Plunge Router
- Straightener bar to help straighten boards while screwing
- Variable 3/8 Cordless Drill
- Drill attachment to apply deck screws w/o you bending
If I had to redo my deck, the one thing I would do differently, is use
those retaining clips which allows one to screw the deck boards from underneath
so there is no visible screws on the deck surface.
But alas they did not have them then :(
And I can guarantee you'll keep using them after the deck is built!
But that is going to put a good dent in that $800 budget. If any is
left, then buy SWMBO a dozen roses and enjoy your new deck!
Good Luck