A network will function at the speed of its slowest component.
As Michael Kjörling noted, a standard cat 5 cable can handle a bandwidth of up to 1 Gbit/s. However, if your internet connection is carried via DSL, then the amount of data being sent by your ISP will be a fraction of this figure. In this case, you will not notice any significant speed increase from using a 'higher' category cable.
If you are using your router to send data between computers or devices on your network, the 'slowest component' rule still applies. You are - in this case - limited to the rate at which device A can send data, the maximum bandwidth of your router and the rate at which device B can receive data, as well as the maximum bandwidths of cables in between. To check the maximum data throughput of each device, compare against the relevant IEEE Ethernet standard (full list here). For wifi devices, check this list.
Lastly, the cat 5e standard brought in controls on crosstalk (where the electrical signals in a wire interfere with other wires which are close by). If your cable happened to be routed near other data wires, then it could be the crosstalk that is slowing down your network. This would be important for offices and such, but for a normal home environment I don't think this would be an issue.
In short, a cat 5 cable is generally good enough!
How fast a connection do you have? Ethernet over Cat5 can support up to 1 Gbit/s speeds although 100 Mbit/s is probably safer to assume. DSL normally operates over phone wiring. Personally, I probably wouldn't worry unless you are actually having problems, in which case you should focus on that, not a particular method of solution (a XY problem question).
– a CVn – 2013-12-11T09:20:36.493