A couple things:
1) Eliminate the rumor mongering in the Union hierarchy -- some opponents hinted at Grant getting drunk even when he clearly wasn't -- they were just trying to open a spot for one of their cronies.
Problem: The chief source of the drunkenness charges beyond the press was none other than G.W. Halleck, Grant's immediate CO, who rather than seeing Grant as a horse to ride got jealous and undercut him at every opportunity, even going so far as to order his arrest and court-martial!

Until Lincoln sent a blistering message to Halleck demanding he immediately "put up or shut up", forcing Halleck to cravenly apologize to Grant for the "terrible miscommunication" and to inform him that the message as his being "relieved" was in fact supposed to represent Grant's promotion to being second-in-command of all Union forces under Halleck.

2) More balanced reporting during the war. Even many of the Northern reporters were biased against Grant because everything was so politicized. They'd report on his victories, but play up any issues and often give out-sized credit to others instead of Grant.
Reporters had their Copperheads too
3) A more successful Grant presidency
That's pushing it. Grant was overwhelmed by the political responsibilities of being POTUS. Among our list of Presidents, he definitely falls under the category of "Innocents", along with (among others) Arthur, Coolidge, Carter, and yes, Wilson (at least in foreign affairs).
4) Reduce/eliminate the lost-cause "histories" that basically deified Lee and suggested that any victory against him were because the odds were so clearly stacked against him.
The Lost Cause histories regarding that are too ingrained in the Southern soul to allow for that. They won't admit being beaten, force to force. EVER. There's even a famous song with a line like that in Ken Burns' Civil War series sung by Waylon Jennings.
5) A good analysis of the facts. a) Grant won and Lee lost, b) Grant's loss rates (taken across all battles) were lower than Lee's and Lee's were doubly bad because the South could ill afford them, c) Grant had a strategy for winning the war, Lee didn't
This last part is the most important. Once Lincoln was re-elected, every last life (on both sides) lost was completely useless. You're not supposed to say this, but this is why I vilify Davis so much. Let the man STAY in Richmond and surrender the Confederate Government, ending the war right then and there, and he gives true closure to the South, lessening the effects of the Lost Cause.
The problem with that? Next to Jubal Early and Nathan Bedford Forrest, Jefferson Davis was the biggest Lost Causer of them all. That's why I can't forgive Andrew Johnson for pardoning Davis. Particularly since Davis neither asked nor wanted any such pardon.
First and foremost, kill the Lost Cause. Lee was a good tactician and a decent strategist, although not perfect, clearly, and that is something that you cannot take away from him. However, he has been deified beyond reason because of the interests of Jubal Early in defaming Longstreet. I don't even think Lee himself really even agreed with most of the Lost Cause ideals- for one thing, he took responsibility for all defeats rather than blaming his subordinates, except for in the case of North Anna, where it legitimately was Ewell's fuckup.
Yes, but without Longstreet Lee no longer had ANY corps commanders that he could rely on. So with Lee stuck in his tent with either angina (or the runs, take your pick), there was no one left to take charge. Just 2nd and 3rd stringers. Even Grant was surprised and grateful that he'd escaped that unintended trap.
However, Lee needs to live longer and get into politics for him to really not be deified.
Having him live longer means he has better health which butterflies every battle in the civil war starting at Chancellorsville.
He was moderate in comparison to many in the South, and might even cooperate with the Republicans, which was Longstreet's greatest sin.
I can't see that. Even if healthier, coming out of the civil war he was E.X.H.A.U.S.T.E.D. He only wanted a quiet life.
If he does this, the Lost Cause and associated movements will not see him in as good of a light.
He'll certainly draw a lot of fire that OTL has been directed at Longstreet. Its been 150+ years since Gettysburg, and STILL Longstreet's statue is only 20 years or so old, with the original shine, AND NO BASE.
As for Grant, he needs to have a better second term. Getting Santo Domingo annexed will help, as he lost prestige over it, and having a cleaner administration of course will also help.
He'd need a staff much better and more honest. Like Warren G. Harding, he trusted his friends and relatives too much.
The rumors of drunkenness were of course bullshit, but there was little he could do about it.
Bullshit? The story was that he never drank when there was work to be done, or when his wife Julia was around. But if so impartial a historian as Shelby Foote considered the stories to be true...
Grant ultimately was screwed because of his association with Sherman, who was vilified by the South, seen as dishonorable by his foes,
War is all Hell

He was the first truly modern general, and Grant knew it.
hated by the War Department,
Who? Stanton?
hated by the Northern Press,
When was this?
and hated by the Indians.
Was there any bluecoat general who wasn't? Mind Sherman & Sheridan were probably in a class by themselves in that department.
Basically, NOBODY LIKED SHERMAN except his troops, the slaves he freed, and Grant himself.
You forgot Congress and the voters up North. After the casualties of Cold Harbor came in, a whispering campaign that had been launched in favor of nominating Grant (who was horrified at the idea) over Lincoln shifted its attentions towards Sherman, until he told them where they could shove that idea.
Grant being the patron of Sherman makes him a huge target. The idea of the duo of the "Madman and the Drunk" was just too juicy to pass up for the press. Grant and Sherman not having such a visible friendship would help him.
Impossible. I can't think of two officers who trusted each other and had a stronger mutual friendship in all the American Civil War than these two men (at their level of command). They wouldn't BE who they were if they were more distant.