When you take off the dead flowers, you need to take off the "dead head" as well, the bulbous bit on the end of the stem. If it's a modern variety this will probably help it to go on and produce more flowers (not necessarily, depends on variety).
It's not the right time of year for pruning, do that in the winter.
There is a lot of rubbish written about pruning: exact time of year, angle of cut, position of cut, etc. Someone did some good trials a few years ago and the results were just as good as the "correct" method when they took a pair of shears and hacked them back. Assuming your rose is not climbing or a standard (tall stem with ball of growth at the top) then sometime between Oct and Feb you should cut it back really hard. You then get a load of strong growth next year and it doesn't spread all over the place.
