I would suggest a good first step is to buy the latest edition of PC Advisor. Has top 10 reviews of PCs in different categories. This will give you an idea of whats needed and you may well find what you are after. the reviews are comprehensive.
With the budget that you should be able to get a PC that has TV output via the PCs graphics card. You do need to check on the spec of the card. Most modern cards will have this. Particularly as you will need a high spec card to run latest games. My advice is to have a look at mag, then contact some of the suppliers and confirm with them.
Usually all you will need is an appropriate cable (s) to connect one to the other. This will probably be into scart on the TV