If you can put together a car stereo, you can put together a computer. It's really not that hard at all. My only recommendation is finding a good local parts warehouse, because piecing things together individually through online ordering is always a huge pain in the ass. It's fine for ordering a key part or two, but sometimes having a good local shop is an absolute necessity.
In Calgary, we've got
http://memoryexpress.com/ . That place is great, though they don't always have everything. I just had to order my 200gb 7200rpm laptop HDD from DirectCanada.com, since MEMX's biggest 7200rpm drive was 160gb. In most cases though, I can find things at MEMX for only a few bucks more than online, without the hassle of shipping to worry about.
For what you need to do, you can go with an absolute bare bones setup. $1000 would probably buy you a computer that is overkill for your needs. For example, if you're not gaming, you can save a huge wad of cash by buying a motherboard with an onboard GPU, instead of buying a fancy aftermarket one. Also, you will definitely not need the latest and greatest CPU, or the fanciest matched pair RAM (gaming computers can be expensive). You can also go with a much smaller PSU, now that you don't have to power all of these things.
For example, something like this would probably be more than enough (copy and paste URL):
http://www.memoryexpress.com/Products/P ... 4(ME).aspx