Is it this one with the Sempron? a low end GPU upgrade might be a decent upgrade option for that machine but frankly it's too outdated to do very much nowadays unless you're thinking of playing some older games.
http://h10025.www1.hp.com/ewfrf/wc/document?cc=us&lc=en...
Amount of VRAM isn't important in the card, it's the actual GPU itself that matters.
If you're on a really tight budget and not thinking of getting a new PC in the near future then you could opt for a low end budget card which should still be an upgrade over your integrated graphics.
You probably wouldn't need a PSU upgrade for a low end card like a GT 720 or R7 240, both of which aren't particularly good values but should save you some money over getting a new PSU.
http://pcpartpicker.com/part/powercolor-video-card-axr7...
http://pcpartpicker.com/part/zotac-video-card-zt7120220...
All in all it won't be a good gaming experience if you're looking to play any recent games.