I have been three times. Campsites are reasonably plentiful and as long as you stick to Late July. August or early September you should be able to get everywhere. It rains a lot and on my last trip we gave up camping after three days and stayed in a hostel type hotel in the interior at Hrauneyjar. This was a very good and comfortable central location which enabled us to explore most of the interior. http://www.hrauneyjar.is/accommodation/
We had the benefit of a serious Icelandic off-road vehicle with 37" tyres lent by a friend and that was only just good enough for some places. There is no rescue service like the AA or RAC in the UK other than Iceland Rescue ( a helicopter service) and you can often wait for 3 or 4 hours for another vehicle to pass by. In one location we drove all day and did not see another vehicle or human being. On another occasion it was obvious that few if any vehicles had actually used that route that summer.
The weather can turn very quickly and it is important if you are intending to walk into the mountains to be well prepared.
Having said all of that Iceland is a fabulously remote place to visit with seriously strange, austere and yet beautiful countryside.
This site was very useful.http://www.typolover.com/_travel/iceland_2009/day_1.htm