I need advice on a decent PC build that will handle game playing while recording yourself via webcam.

My daughter loves PC gaming and likes to record herself at the same time for posting to YouTube. However, on her old PC (HP a1730n with ATI Radeon HD 4600 and 3GB RAM), doing both things at the same time results in choppy playback. I've built a couple of PCs before for home office purposes, but never for gaming. I thought it would be a fun project for the two of us, but the options are dizzying, and I'm on a budget (maybe $500 max). Any advice would be appreciated. Thanks!