I've done C25K in the past, I think it's a great program! Once the weather improves a bit more, and there are at least 3 above-freezing days each week, I would like to get back into it. I've been out running a few times in the last month, and I'm probably at about a week 4 or 5 level. There are almost always threads in the exercise forum for people doing this plan to support each other, you should look around.
As for speed, you should feel like it's challenging, but like you could still carry on a conversation if you wanted. I don't recommend doing this in a gym

but if you run at home, try talking to yourself to see if you're pushing too hard. If you wanted to be able to run the 5K in 30 minutes, you're looking at around 6.0mph.
Keep in mind that running on a treadmill doesn't always translate well into running on a track or outside or whatever. I've heard some people recommend that you put the treadmill at a 1-2% incline to simulate the extra effort it takes to actually move yourself forward. So if you start running outdoors (which I recommend at least doing at least a few times before running an outdoor race) you might find that you're slower.