As I understand it (I'm not a pilot), radio communications are a more-or-less first come first served proposition. If I'm in busy airspace with near constant communication between ATC & aircraft, how do I break in to make contact with the ATC for whatever purpose necessary (establishing comms to enter airspace, or whatever the situation may be)?
As Ron commented, the basic answer is that you try to anticipate when a short gap is going to be, and jump on before someone else does. It sounds a bit daunting, but it actually works out pretty well in practice.
That being said, it's not exactly first come first serve. ATC tends to prioritize talking to the aircraft which need the most immediate instructions. So, an aircraft on final is going to get more priority than one which hasn't even entered the airspace yet. Even if you manage to get your request in, if the controller has a more immediate issue to address, you may get completely ignored. Sometimes they'll come back with "calling approach, say again" or sometimes you just have to try again later.
Either way, it's usually not too difficult to stay out of trouble long enough to get someone's attention. If you're IFR or have already made initial contact, you can probably just continue on your path and you'll eventually become a big enough priority for them that they'll call you if you still haven't a chance to get a word in. If you're VFR and haven't made initial contact, you can probably just avoid the airspace for a few minutes.
Key up the mic and state:
If the frequency is really busy, some people will substitute #3, 4, 5 with "request" (e.g. "Seattle Approach, Skyhawk 1234, VFR request"). Once the controller responds the remaining details can be filled in.
The first thing to get clear is: who are we talking to here? a tower? an ARTCC? what?
In my experience the biggest problem can be trying to reach a busy tower for a small or medium-sized airport. In other situations, usually ATC calls you, not the other way around. For example, if you request vectoring, you only have to do it once; after that they call you.
If you are trying to reach a tower and they are tied up, it can be a real problem because as you get closer to the airport you need an answer. In one or two cases I have actually had to circle to wait for some chat head to shut up so I could talk to the tower and get instructions for an approach.
Commercial flights have this problem less because they are operating on filed flight plans, so the tower is ready and waiting for them.
Another serious problem is when you are in the pattern on downwind and the tower is too busy to give you a landing clearance. In this situation all you can do is to keep flying downwind. Sometimes it gets ridiculous. I have flown literally miles downwind waiting for clearance to turn onto final, which then turns into this loooooong 5 mile final.
If you fly downwind too far, you might even have to leave the pattern completely and start the whole process all over again, although this has never happened to me.