Once you graduate from a UK university, your student visa will expire and you'll have to return to the US. Your time here in the UK studying won't give you any additional benefits when it comes to seeking employment or being eligible to work in the UK.
You'll have to go back to the US and seek employment from there which is usually fairly difficult. As expat said above though, your best bet is to study something that is in high demand here in the UK, go back to the US after your studying is completed, and apply for an HMSP visa. I'm afraid once you graduate (unless you happen to be studying in Scotland?) you won't have any more rights to work here in the UK than any other American.