This is an interesting and broad question. The closest I could find to caniuse for TV's, was html5test.com. So here is a good place to start: http://html5test.com/results/television.html
As far as I know, most newer TV web browsers are based on WebKit, however some have certain restrictions, such as local storage.
If you know which TV you'll be developing for, you're probably best off looking at the support of that particular brands chosen browser.
Here's a quick compilation of links for your convenience:
Based on the results from html5test.com, any TV with the Evo Browser (or probably also the Opera browser) currently seems to be the best choice when looking at HTML5 support. In general, you probably can't go wrong if you go for a TV with either of those browsers. The Evo Browser seems to be gaining some traction, having partnered up with several large brands. Opera however has some seemingly nice developer tools (emulator), so it might be the better choice.
If you don't know what you'll be developing for, you're generally best off going for jQuery, but you could also read up on Feature Detection in order to provide the best possible user experience for your web application on any given TV.
Here is a list of possible alternatives to jQuery:
They're not specifically for TV browsers, so jQuery might still be the better choice.
As an entirely different solution, you could just buy a cheap Raspberry Pi, load it up with your linux distro of choice, and hook it up to the TV. Then you can use whichever desktop browser you prefer, with all its glorious features.