How do I figure out lumber amounts for a second story deck off the back of my house?

The way most people do the material list is they don't, actually. You draw the thing out, even just a sketch with dimensions, and take it to the lumber yard counter. Any yard worth buying from will do a takeoff for you, meaning their guy figures the joists, beams, posts, hardware, everything, and quotes it as a package. I did that on my brother's walkout deck and it saved me from ordering wrong twice. If you want to ballpark it yourself first, sketch the frame on graph paper, lay out your joists at 16 inches on center along the length, count beams and posts from your spans, then let the yard correct you. The fasteners and hardware alone will surprise you, budget real money there.

One thing you relly can't shortcut on a second story deck is the ledger and the bracing. That connection back to the house and the diagonal bracing on the posts is what keeps the whole thing from racking or pulling off the wall. Your local building department will want a permit and drawings for anything elevated like this, and honestly the permit process helps you here because they'll catch undersized posts and footing sizes before you pour concrete. The frost line comment above is dead right too, everything starts below it or you'll be shimming a heaved deck in two years.

On the DIY question, I'll be honest. Framing on the ground and standing posts is very doable for a handy person. But you'll spend real time working off ladders and scaffolding twenty feet up with big sticks of lumber, and that's the part that gets people hurt. If you've never set a beam with a couple helpers, plan on renting the right equipment and having at least one experienced friend there for the heavy days. Lots of guys do exactly what you're thinking, DIY everything up through framing, then bring in an electrician for the post lights and maybe a pro for the roof structure if you do the covered part. For learning, search for full build series on YouTube rather than single videos. There are a few channels that document an entire elevated deck start to finish, footings to railings, and watching the whole sequence makes the plan click. Also grab the deck guide off your local code office website, most of them publish one and it's basially a free step by step.

I'm in the early planning stage of almost the same thing, mine would sit over a walkout basement so I keep thinking about the space below. Does anyone know how you keep rain from dripping through the boards onto whatever is underneath? I've seen pictures of plastic-looking panels and channels that go between the bottom of the deck frame and the floor joists but I have no idea what that whole setup is called or if it's something a beginner can install after the deck is done. Seems like it would be way easier to figure out now than after everything is built, but maybe that's wrong.

One thing to figure out early is how the deck attaches to a brick house. If that's brick veneer and not structural brick, you can't lag a ledger into it, the fasteners have no real holding power. On my place in the 90s the old-timers just set the deck freestanding, posts a foot off the house, and honestly that's still a clean way to do it. Costs you two more footings maybe, but the deck does its own work. Worth asking whoever checks your plans which way they want it before you buy lumber.

I was in a similar spot last year, except mine was a low grade-level deck off a ranch house. Even three feet up I underestimated how long the layout and digging stages take, and I had it easy on height. If I were doing yours, I'd prolly do everything on the ground myself, then hire just the beam and post lift day, even though I said I'd never hire anyone. Not sure what that costs around you though, is that the kind of thing a framing crew will do as a one-off or do they want the whole job?