PC build, is this possible?

If you want to split sound to watch two different videos on the two TVs from the same computer, you can run two instances of VLC and manually select the audio source for each.

In order to do this, you actually need two audio sources. Otherwise you can only send the same sound to both TVs. One way to do this is to buy a cheap dedicated video card like the GT 610 or R5-230, and build an intel system (integrated GPU) using a motherboard with HDMI out. One TV would be connected via HDMI to your dedicated card, and the other via HDMI to your motherboard -- the former uses the video card's HDMI audio source, and the latter will use your motherboard's onboard audio. This will only work on software that's intelligent enough to allow you to select audio sources. You'll have to manually select the sources each time (or there may be software that can make it more automated. Haven't messed with that). This also may work with an AMD APU build, but I am not 100% sure.

This is the most basic of solutions, but your question is "can I utilize both tvs while running separate programs simultaneously?" -- If that means two people actively using the same PC at the same time (e.g., one person browsing the web and someone else playing a game), you'll need additional third-party software. Windows MultiPoint is a solution I've seen used in schools. Others may be more familiar with better solutions. I'd probably just use VMware.