LR31 Captain experiences a cabin pressure warning light after a rushed takeoff and levels off at 9;000 FT to troubleshoot. Communication difficulties are also reported both between the pilots and with ATC resulting in squawking 7600. Crew is vectored for an overweight landing.

Narrative

We received ATIS; said to expect Runway XXR. When we taxied out we were sent to Runway YYL; took off YYL. Rushed behind a landing Airbus and a Boeing on 4 mile final. Climbing through 8;700 FT we receive a cabin altitude light. The Captain leveled the aircraft at 9;000 breaking through 9;500 on the level off. We were cleared to FL210. We were climbing over 8;000 FPM. The Captain called Departure Control and said we had a potential issue and that we needed to stop the climb at 9;000 FT. After inspection we found the cabin altitude to be that of our altitude. We then were given a turn north 360. The emergency pressurization turned on at 9;500 FT and rushed a large amount of air into the cabin. We were then given a turn on a downwind and a descent to 6;000 FT which we did. At this time our right fuel computer kicked off line; which took approximately 1.5 minutes from our attention. We reset the computer and it came back online. Departure was very busy with arrivals at this time and missed for over 4 minutes that we never broadcast a reply. They then called us repeatedly; which we responded too; however our microphones were not transmitting; they were also not allowing the crew members to talk to one another or hear each other. We set 7600 in our transponder. We could still hear ATC. We tried all backup radios with no luck. ATC then vectored us southbound over final and looped us back around to our final by asking us to ident if we could comply; which we did. We then were cleared to land on Runway ZZ behind a Learjet. We taxied off behind the Learjet and thankfully he was going to the same FBO we came from so ATC just told us to follow him. In the 20 minute flight a lot happened. Mainly because of the radios; no crew communication and being at such a busy airport; we didn't realize that we landed approximately 75-100 LBS overweight. I am not even sure how I could have landed back into the airport and told ATC that I needed delaying vectors to burn fuel considering how busy it was with no radios. So we landed.