What will border control need from UK citizen visiting the US?

If he has his ESTA approval and his passport, all he really might really need to do is show that he enough money to support himself for the amount of time he is visiting. It would also help if he could show evidence of where he is staying - because they might ask about it. You might want to be there at the airport to meet him - just in case they wish to question you. :-)

Your boyfriend (yes, I´m reading between the lines here) needs to show that he can financially support himself for his entire vacation. Depending on where he will be living, hotels or with friends (girlfriend) I would say he would need between $1000 - $2000 per month, but thats my guess, the CBP officer might see it differently.

He might not have any problems at all, and he might be questioned about his intentions, his financing, his ties to the UK etc. He needs to be able to prove that he plans to return to the UK, to a job, studies, apartment he pays rent for, family.

If they suspect that he plans to immigrate, then he will not be allowed into the US.

All that being said, I dont think he will have any problems, but it is best to be prepared.

If you plan to get married there´s a lot of information here:

http://familybasedimmigration.com/

Your best source of information would be the American embassy in the U.K. They will be happy to advise him on how to get his paperwork and luggage in proper order so that there will be no delays in customs (border control) The United States does not want to detain or delay anyone who is coming in to America legally.

I'm thinking your visa and your passport should be sufficient.