You can, it will take about 30-45 mins and will cost a fortune.
You could catch the bus, much cheaper. Buses run from outside Stoke Station (although not directly to Towers) so catch the bus into Hanley Bus Station (about 1 1/2 miles away) and from there you can get a bus to Alton Towers - the service is 32A. The X39 used to run directly from the Station to Alton Towers but I'm not sure if this service stil operates.
Or if you are travelling on Virgin, they run a bus from the station directly to Alton Towers, see
here or
here