I'll be frank...the answer is quite simply no. The resolution for an SD tv is simply way too low for a modern computer to output correctly - and even if it did, it'd be incrediblely blurry.
Let me explain further, the average screen resolution that the consumer has is 1024 x 768, a signal for broadcast on an SD TV is at most 720 x 576, and there is no display option on any modern computer than goes below 800 x 600, and that's at a stretch. This, sadly, is the truth - so that "crap, cheap monitor" is frankly your only option. That being said there are plenty of passable (note: PASSABLE) monitors being sold for less than £100 these days, and of course, there's always ebay.
If you've such a small budget that you can't build a system without resorting to a SD TV, I suggest just plumping for a cheap and cheerful pre-built system, otherwise its more of a problem than its worth.