What is the easiest way to purchase a firearm from a private seller and give said firearm as a gift and still get that look of surprise when they open it. Basically hoping there is a way to forego two sets of paperwork.
Depends entirely upon the state of residence of all the parties involved: seller, purchaser and recipient of the gift. For example, if all the parties involved were residents of Washington state there would be no paperwork involved at all. You, as a Washington resident, would purchase the firearm from another Washington resident, wrap the gun in a box and put it under the tree and the Washington resident recipient would unwrap the gun in the box just like any other gift.
Or, if you, as a Washington state resident, chose to purchase the firearm from a dealer, in Washington state, you would do the form 4473 at the dealer, get the NICS background check done on you at the dealer, take the gun home, wrap it in a box, put it under the tree, and the Washington state resident resident would unwrap the gun in the box just like any other gift.
Other states may be more complicated between same state residents.
If all parties are not residents of the same state then Federal law requires your gift to go through an FFL. You could still wrap the gun and put it under the tree, but you would have to be physically present when it was opened and after they opened it, then you would have to deliver the gun to an FFL for formal transfer to them.