In reply to JamButty:
Running up hills isn't easy, so it's important to drop down the gears and not try and run up to fast. Quite often fast uphill walking is more efficient.
If you are struggling to keep going not just on the uphills then this sounds like you just need to get used to the nature of the running and that the rhythm is different to trails due to the roughness of it.
In order to make this easier to get to grips with though you can improved you recover time by doing interval training. I prefer hill intervals as these build the muscles also. A set could involve running up and jogging down a steep hill for two pyramids of 15, 30, 60, 90,60,30,15 seconds with a bit of a break in between. These should be pushing hard, and should probably make you feel a bit ill! But do them, or other variations for a few weeks, and you will certainly notice the difference!
Good luck with the running!