Using that TV is very sensible. It may be a lower res that you'd like but it does make life easier on the GPU. HDMI from the GPU will also provide sound to the TV saving you buying speakers. (if the TV has an RGB or DVI port, using this will probably get better results but you'll need a cable for the audio or some speakers).
THe 6870 you've chosen isn't very good to be honest. Not reallly any better than the 5870. It's priced the same as some of the 6950 cards which would be a much better card for gaming. *EDIT* Just seen your spec and prices. The card is cheaper but not by much, still worth the upgrade *EDIT*
You really need to decide now if you will ever use more than one gfx card. If you want these upgrade path it will add costs. You will need a higher rated PSU and also the mobo to support it properly (both PCI-E lanes in sync).
Good luck with the build. Any Qs or problems when you build you know where we are.