TRACON Controller described an unauthorized climb by an IFR departure off a satellite airport; the reporter listing the required use of enroute RADAR information as contributory.

Narrative

I was training a developmental on satellite north/south combined; a PRM1 was issued an IFR release off ZZZ1 230 heading and 3;000 FT. The PRM1 checked on frequency on the 230 heading and 3;000 FT; the PRM1 was RADAR identified. A FA50 was issued a climb to 13;000 which the FA50 acknowledge and we made a land line coordination; at this time it was observed the PRM1 was climbing out of 4;700. The PRM1 was instructed to maintain 6;000; he kept climbing he was instructed to maintain 7;000 FT which he finally acknowledge but said he was through 7;400 and would descend to 7;000 FT. We have to work satellite position on a secondary long range RADAR from Center; since the airspace we work is more then 60 miles from a major airport and anything below 5;000 we cannot see on the airport RADAR. When the PRM1 started climbing on his own; although not sure if he took the FA50 climb; he never acknowledged a climb; we saw he was out of 4;700 FT as we were instructing him to maintain 6;000 FT his altitude indicated 5;400 and before he could answer us the next read out said 6;700 FT then we said maintain 7;000 FT which he again went through and stopped at 7;400 FT and descended. This is due to the lag time from the RADAR to our RADAR screens as you can see very dangerous. My recommendation is [to] redesign airspace so we will be working within 60 miles and not have to work on a secondary RADAR site eliminating a lag on altitude.