Yes, you can. Your license is tied to your Battlenet account and lets you install D2R on as many of your PCs as you want. You access that license by logging in through your Bnet account. You can only play one at a time though.
Also keep in mind, if you play single player offline, you will want to put your chars on a memory stick or something.
Online chars will always be there regardless of the PC you use to log in. They are saved server side.
Not saying you were planning on it, but do keep in mind account sharing is not allowed (except one minor child you are the legal guardian of).