I have a friend who closed on his first house on June 15, 2010 and actually started the loan process in mid April. He is in the process of trying to amend his taxes to get the credit now, but what proof would he need to prove that he was in the process of buying before the April 30th deadline?

" but what proof would he need to prove that he was in the process of buying before the April 30th deadline?"---> Your freind must meet the required deadlines to be eligible to claim the credit. Your freind must have bought or entered into a binding contract to buy a principal residence on or before April 30, 2010( I guess he'd actually entered into a binding contract before April 30, 2010; as you said that he started his loan process in mid-April).
If he entered into a binding contract by April 30, 2010, then he must close (go to settlement) on the home on or before Sept. 30, 2010, however, he actually closed on it on Jun 15 2010.So,he must use Form 1040 to claim the credit; he cannot use Form 1040-EZ or 1040-A.
Also he must attach documentation,i.e., contract or escrow record or names and signatures (if available) of all parties involved, Property address,Purchase price, and date of purchaseetc. showing the purchase of a home between the applicable dates.