Alternators need the motor to be running at about 1000 RPM for their maximum (best & efficiet) output, & it shouyld be around the 14V mark (13.8V is optimal). It sounds like your alternator is working fine.
To check the battery, put it on a charger over night. Then check each cell with an acid level meter (I don't know it's correct name. Its a glass tube with a float inside it with a rubber bulb on the top). Each cell of the bettery should come up to the same float level, ideally in the green band on the float.