What are three of the things Lee hoped to accomplish by invading the North?

Answer 1

Through both the 1862 and 1863 invasions of the North, General Lee led his Confederate troops into Northern territory for numerous reasons. Perhaps most importantly, he intended to win such a decisive victory that the North would sue for peace or that the South might win formal recognition (and even material support) from European nations. And yet, the invasion was also motivated by the desire to take the battle into Northern lands, thereby helping Virginia landowners to recuperate from earlier conflicts in the war and perhaps even relieving pressure on the Confederate forces in the Western Theater, who had suffered a number of defeats at the hands of the Union forces active there.

Answer 2

First time round (Sept. 1862), he was wanting to assert Confederate viability in the eyes of the British and French, who were close to sending military aid.

Second time round (June 1863), he was wanting to relieve pressure on Vicksburg by provoking Grant into sending troops East to counter the threat of an invasion of Pennsylvania. By this time, the South was feeling its lack of supplies, and Lee was hoping to do some foraging, as well as occupying a shoe-factory near Gettysburg for the benefit of his barefoot and hungry troops.