Heathrow airport is miles from central London. Its pretty far from the suburbs which are already a good hour from central London by public transport.
To live alone anywhere decent in London will cost over £1,000 a month plus all bills so add another £200 easy.
You will have to share (unless earning mad scrilla...) There is nothing good near Heathrow, I used to work near there once!
You can find somewhere to live near the Piccadilly Line (London Underground/the tube) - look on a map. It's the only tube line that serves Heathrow and connects to central London quite quickly.
You could for example, find a place in Kings Cross. A walk to central London so no need for night buses. Good access to East London where the best bars are. It's Zone 1 so very central. A bit grubby and used to be where all the whores and crack heads hung out but is much better now after a clean up and regen.
Then you can just take the tube to work every day on the Piccadilly Line. But we are talking an hour+ on the train to get to work. (anywhere in London takes about 45 mins even if just a bus down the road)
Or live somewhere West London like Acton/Ealing which is shit but will have a few local bars that will be shit but will be cheaper to live. Then you could just got out at weekends and get taxi/night buses home.
Depends what your priority is - ease of getting to work or having access to good nightspots - in your situation, you can't have both! Sorry.
This is a good website for finding places to live/share in London. You'll get an idea of prices/areas. It is quite expensive but so is London:
http://www.moveflat.com
For cheaper, check out
http://www.gumtree.com. You can find bargains but a lot of shitholes too and the site is crap to use compared with the layout of moveflat.com
Hope that was not too depressing!