In the understanding that what is meant here is a campaign up the Peninsula (York and James) rather than a move south to the Roanoke or similar, here's my evaluation.
1) Available force for the attack, and for the defence of Washington.
The strength with which Grant initially crossed the Rapidan is around 140,000 PFD. In addition, the active part of the Army of the James (Yorktown, Gloucester and Portsmouth, the latter being the cavalry and siege artillery) was around 37,500.
This gives a strength pool of over 170,000 regulation PFD to work with.
Assuming no further reinforcements and that the historical reinforcements instead go to Washington, then the available strength for the defence of Washington is rather noticeably over 50,000 PFD (which is the scale of reinforcements sent to Grant, plus the strength historically left in Washington, plus what was in the Valley).
The force available for the attack that forms the Army of the Potomac could not all be moved in one lift. I would suggest moving one cavalry division, then moving one infantry corps at a time, and finishing up with the rest of the cavalry required on the Peninsula.
2) Operational barriers.
The York river is open, and the James river is open at least as far up as City Point. The Yorktown line is gone, as is the Williamsport line; the campaign could be fought by marching up the Peninsula and supplying from the Richmond and York railroad or by supplying from the James (either side of the river).
The only defensive line that Lee can adopt which Grant could not easily turn is the Richmond defences themselves. (The Chickahominy can be turned by a landing south of the James, and Grant would have enough riverine mobility to move at least a corps at a time.)
3) Overcoming the Richmond defences.
Historically speaking regular approaches along the Richmond and York were viable with enough troops to cover the flank of the army, and they would be viable in this counterfactual (140,000 between the Pamunkey and White Oak with 35,000 at Bermuda Hundred gives Lee some very difficult choices). There are alternatives as well, of course, such as advancing up both banks of the James or similar.
4) Grant's command pattern.
Grant's command pattern of assaulting works (which he had in the original Overland) might emerge here as well, and would be the largest possible problem with the method. The Richmond defences were strong.
5) Movement to the Peninsula.
It takes longer for Lee to move troops south than it would for an already loaded corps to reach the Peninsula and unload, though the complete load-transit-unload-return cycle for a corps would probably take longer than Lee's reaction time. Lee however can't just send south a corps in reaction to each Union corps, or he'd end up with a single Confederate corps potentially facing two full Union ones with ample cavalry support.