Cat.5 doesn't exist any more, it's just Cat.5E today. That cabling will be fine for up to gigabit Ethernet (as will Cat.5).
Keep in mind that a network connection is very versatile. Depending on the where you want to run the cabling you don't necessarily need to run all wires from a central point. You can run the router connection up to the living room with a single cable, put a network switch there and connect everthing else to that. As long as you've got enough throughput you can run as many 'connections' through a single cable. Most people are still fine with fast Ethernet (100 Mbps), but gigabit is nicer if you want to run media servers and such while still being very affordable.
I started out connecting everything to my router but one day 100 Mbps wasn't fast enough any more, so its integrated switch was complemented with a gigabit one. When the TV and Wii were added I ran a single cable to the living room and put a second switch there, connecting those, another PC and a laptop to the rest.
