~Nobody can any better than Robert E. Lee and his Commander-in-Chief, Jefferson Davis. Look them up and see what they had to say about the Gettysburg Campaign. There was a heckuva lot more involved than a single battle, so I suggest you focus your research on the Gettysburg Campaign, not on just a few days in July, 1863.
If you don’t want to do something that simple, then just go with:
He wanted to win the war and procure a treaty which would recognize the independence of The Confederate States of America.
His purpose was to bring the war to the North and to cause enough popular discontent to compel Union leadership to abandon the war and sue for peace. He was quite aware that he probably could not actually capture Washington and President Lincoln, but he hoped that fear and an increasing financial burden would be enough to force at least a compromise which would satisfy his southern compatriots. As you know, it didn’t work, and the war dragged on for another two years.
Bobby Lee wanted to hit the North in their own backyard rather than fight on Southern soil to show the war was not going to be only defensive and he also was looking for a shoe factory in the Gettysburg area so he could get footwear for his soldiers.
A great novel which describes the Battles at Gettysburg is Killer Angels by Michael Shaara. You are there!
1. To help relieve the siege of Vicksburg the lost of which would have been (and was) a massive blow to the Confederacy. This was the MAIN reason for the whole campaign.
2. To attempt show Britain and France that the Confederacy was capable of beating the Army of the Potomac on northern soil and not just in defensive engagements. This (he hoped) would demonstrate that the Confederacy was capable of sustaining itself as a independent nation.
3. To relive northern Virginia of the burden of the war in the east, so that the land could be worked and food grown again.
4. To obtain badly needed food, supplies and, yes, shoes.
Without Grant’s siege of Vicksburg though, Gettysburg would never have taken place.