Here are your options:
1) use adapter for vga to HDMI i/f
2) install video card(likely pci type)
3) get a new PC with latest i/f
The ability to add new video card gives the benefit of returning on-board video ram use back to system. Of course, the new card will have its own video ram and should have the ports to accept either VGA or HDMI(verify), one or both. Your HP installed Intel chipset should allow the added video upgrade path. You will need to disable on-board video or follow any install instructions in order to use new video card. Understand if youy get a truly hi-powered video card, that your HP PSU be capable to handle it. Thus, the consideration of yet getting a new hi-wattage PSU maybe required. This will allow and maintain the system uptime for now and later. That's important, should you may buy a new video card and for the time being works great, then later the system seems to not boot or runs some boo-boo all due to weak and faltering PSU.
I have a 3+ year old "off the shelf" HP a6120n desktop
http://tinyurl.com/6nkb86 <- link to specs....
running Vista (loathe Windows 7 for many reasons), maxed out at 4 gigs of ddr2 ram and, of course, integrated graphics.
At one point I got the bright idea to upgrade to a newer HP computer and also bought a new, Samsung P2350 monitor with HDMI capability. When the new computer crashed 20+ times in 4 days, I returned it but kept the monitor. Now I have the bright idea to find out if I can upgrade my video capabilities incorporating the HDMI quality of my monitor. Is this even possible? Am I, as the saying goes, trying to "put a silver saddle on a mule"?
My post clearly shows I know enough to get myself into trouble.