Chris, I think the idea is feasible and straightforward. The one negative, and I don't think it is a showstopper, is that the level coming out of RSSI will not be anywhere near a perfect indicator of exact range from the transmitter. You see, the RF has a tendency to bounce around a bit and as mentioned, the signal strength will also be strongly influenced by the orientation of each player's body. But I don't think this is a huge negative. In fact, it could add more complexity to the game and make things a little bit less predictable, which could be more interesting and perhaps add to the player's experience. The player would have to learn the personality of the tones that he hears and all of the quirks involved too. I think it could be quite cool actually.
You are wise to start with pre-made receivers and transmitters. Building these from scratch is beyond the scope of what you are trying to do. You have the right idea with the RSSI output. These are very linear with signal strength in dB and you should be able to put these into the PIC A/D converter without trouble. RSSI voltage will vary fairly quickly in tracking the signal level. But the RSSI can also be a bit noisy, that is, have higher audio frequency noise on the DC. So you should plan to include a simple low pass filter with a cutoff frequency high enough to let the level track changes due to fast movement, but low enough to get rid of as much noise as possible. Perhaps 1 KHz would be a good starting point as this will allow a slew rate on the order of about 1V per milliscond. A simple RC filter should be enough.
The antennas will be key to range and consistency around the player. You may want to consider the tradeoffs. On the one hand, at 430 MHz, the antenna should probably be at least 100mm long or so and preferably a bit longer. You can use off-the-shelf antennas, but most of them work best "in the clear", that is when not laid flat against the body. Can you consider a mounting method where the antenna is up on the shoulder or on the head of the player? That would be ideal.