I got accepted in one of the top-tier school in US for PhD program in Engineering however my dream is to go to different graduate school. I am planning to send a follow up email to one my potential advisor. Can you please tell me how to start and should I mention that I got in to PhD program in one the top-tier school? Should I start with as we talked before or as I mentioned in previous email?
No. You will appear to be threatening to go elsewhere if they do not act on your application soon. People may react to such (perceived) threats by saying, "Don't let the door hit you on the way out."
If the other institution has a deadline by which you must accept their offer, tell your chosen school, "I really want to come to [chosen school] but other factors require me to make a decision by [week before deadline date.] Is it possible you could help with that?"