Is there a way to allow multiple PCs behind a router to be able to use a range of ports (for BitTorrent)?
There's probably a better way to say this.
I don't have a good understanding of how BitTorrent works, but from your question I am going to assume that like any other server, BitTorrent listens on (a) specified port(s) and accepts incoming connections to the port(s), much like a webserver listens on port 80 and will accept connections from outside on that port. And having two or more computers behind a router listening on the same port will cause a conflict. If this is indeed the case, then there are three answers to your question:
1) It is not possible.
2) It is possible, if BitTorrent allows using nonstandard ports or multiple standard ports to have the software listening on. Perhaps you can setup each computer listening on different ports (computer foo listens on port 1, computer bar listens on port 2, etc.). This also depends on the configurability of the BitTorrent software, which I am not familiar with. But from the wording of you question, it seems you want the same range of ports allowing incoming connections on all computers on the network.
3) It is possible, depending on the friendliness of your Internet Service Provider and/or the Network Address Translation (NAT) configuration options on your router. In this answer to your question, there are two sub-answers:
b) Ask your ISP to setup your connection as a bridge, causing your router to act as a hub rather than a router. Your ISP then assigns public IP addresses to each computer on the network. This might not be a good option since your router no longer can act as a firewall since it is transparent and has no IP address.
Here is a reference, but it doesn't really include a whole lot more information than this post: