The speed it not determined by the cable it is determined by the ports in the equipment. If you have 100m ports it will go 100m , if you have 1g ports it will go 1g, if you have 10g ports it will go 10g. So if you were to put a cable rated to run at 10g into a 100m port it will still only run at 100m.
Pretty much any cable you find will run at 100m. You should use cat5e if you have gig ports in your equipment. Since I doubt you spent money to have a router with 10g ports no faster cable is needed.
Now you might actually find cat6 cable cheaper than cat5e just because of marketing. Things like the flexibility of the cable, the types of ends, the color etc are more the differences. Pretty much any cable you will find for sale will be able to run 1g.
That said if your router ports are only 100m then the cable you have now likely will have no trouble running at that speed. It is highly likely the cable you have now is cat5e or better since nobody sold less than cat5e for many years. It will likely be marked on the side of the cable.