The "Lee Strategy" for winning the war, to the extent that he had one, essentially was to inflict a Cannae-level defeat on the Army of the Potomac in the hopes that this would frighten the Lincoln administration into seeking peace talks. It was not realistic and wouldn't have worked, but that was how he thought.
Lee went into Pennsylvania largely for logistical reasons, to give Virginia a break and supply his army by foraging in Union territory. But it was also to force the Army of the Potomac to respond to his movements so that he could seek battle under circumstances of his own choosing. I think Lee understood that he could not take Washington City. His target was the enemy army, not any enemy city.
Moreover, as he indicated in a letter to Jefferson Davis on June 10, 1863, Lee thought the South needed to be encouraging the peace movement in the Union (a rare and unsolicited bit of political advice, which shows how important Lee thought it was). Although he didn't mention the movement into Pennsylvania in his letter, the fact that Lee was thinking about it so much at the time suggests that he believed a Confederate victory on Northern soil would dishearten the hawks in the Union and encourage those who were beginning to push for peace.