The Sempron is AMD's competing product to Intel's Celeron, AFAIK. In theory, you'd want a CPU with at least 2 or 3 cores, and video RAM of at least 256 MB (meaning if you're lucky enough to have an onboard graphics card with very stable 3D acceleration, it might suffice). Personally I'd recommend getting an Athlon, it's probably much better than a Sempron, and it's also obsolete so prices have been dropping phenomenally (the Phenom is still somewhat expensive, but I gather you want the cheapest price). I just bought a desktop kit that came with a Athlon 3 core CPU, it cost me $235 for the computer (after taxes, shipping and mail-in rebate deduction), granted that didn't include the monitor, mouse & keyboard, and I also got a dedicated graphics card, but I was able to get those at competitive prices. A Sempron CPU might do the trick, seems like the graphics card is more important than the CPU for getting 1080p, but considering the price difference I'd have to recommend the Athlon as much better value.
Read more here: http://geeknizer.com/1080p-minimum-requirements/ (article is a few years old, but still relevant to your question).