So what is it you want to do in the end?
From TV to computer would imply you want to record the TV you watch on your computer and in that case you would still need a TV tuner card (don't know of any that accept HDMI / DVI / VGA though) to provide an input and this is presuming your TV has an output (a new LCD surely would).
Or if you mean computer outputting DVI and your TV only has HDMI input then there may be an adapter but HDMI is meant for video + sound in 1 cable so the adapter would need to also plug into your digital sound out from your mobo / sound card.
In general I guess you are going for one monitor to do both computing and watching TV, so you'd have to decide what you'll mainly be using it for:
Almost exclusively TV with the computing only being games and light internet sessions (no extended periods of time): get an LCD TV ~27" for your budget.
Homework, lots of email, spending time on internet: get a 19-20" Computer Monitor (sign up for Dell newsletter as they always have good deals on these) and a TV tuner card for your PC. The tuner will let you watch TV on your PC and record from VCR.
Jo