I'd like to clear up a little misconception about Grant. He wasn't drunk all the time. He had migraines.
Now, onto the question...
In the beginning, most of the talented generals came from the South. Most of those chose to believe that their individual states were more important than the United States as a whole, and so joined the CSA. Sectionalism (North, South, West regions, rather than Nationalism; one country) was quite rampant by then. The Missouri Compromise and Kansas-Nebraska Act (and the evils the KNA gave birth to) were indicators that something wasn't right with the USA.
The North believed the South was going to be a pushover. They had half as many people, little industry, and a lost cause. What more could one want of an enemy? Lee took the initiative and kept a hold of it. By 1862, when Lee was in Pennsylvania, trying to feint his way into DC, it was looking highly likely that the South would win. The North couldn't win battles. McClellan was a political opponent of Abe Lincoln and wanted to be the next President. He was also pretty incompetent. He wouldn't attack; he usually believed he was far outnumbered, and Lincoln couldn't can him until he did something that even Congress could see was wrong. England and France were considering helping the CSA.
Then a Confederate courier lost Lee's orders. They were found by the Union and Lee was "handily" defeated; driven out of Pennsylvania. Lincoln was then able to issue the Emancipation Proclamation, freeing slaves in the South. He had to wait a long time till the Union got a significant victory. By issuing the Proclamation, he guaranteed that European powers (who detested slavery by now) would not aid the South, sealing their downfall.
Now that Lee had lost the initiative, the Union was able to press their advantages that they had been so proud of. The Union had rail lines more thoroughly distributed, allowing men and material to quickly be taken where they were needed. The South was suffering invasions, having transportation and production eliminated, and it became an easy matter for the North to concentrate all their power on small chunks of the South, which couldn't be reinforced.
Out west, Grant was having successes. He captured Vicksburg. He made promises to Lincoln about what he would do, and he kept them. By the time he was made General of the Armies, the South was going to lose. He probably shortened the war by a year or two. The South would likely have tried to keep on fighting, had he not quickly ground Lee's forces into the dirt.
If you play conquer-the-world or conquer-the-universe games, you will likely see that he who has the greatest production generally wins the war. Of course, now we keep adding lots of new elements to the mix, like technology, national morale, and other stuff, but in this war, the North had most of the advantages, and eventually won. If you look at our other wars, you'll notice a pattern. Generally, we get caught by surprise. We lose. But then we kick our production into gear, and we kick butt. CW, WW1, WW2, KW (twice), and Cold War. That's right. We did a massive buildup and the Soviets couldn't match us, so they fell.