Lee never wrote his memoirs, and as such his motives seem to have been somewhat veiled by second hand accounts, etc. Is there any definitive information on his primary reasoning and end goal(s) for invading?
Lee's reasoning for going North, while clouded at the time by second hand accounts (and second guessing) are easily identified by the communication back and forth between himself and Richmond. That reasoning was
To follow up on the momentum of the stunning Chancellorsville victory. Lee believed that the North and Army of the Potomac were teetering and a simple push or show of force backed by a decisive victory or several small engagements that beat a cowed Northern army would convince the Lincoln administration to the negotiating table or at least greatly weaken Lincoln's position in next year's election.
To generate international support primarily French and British support. Both Lee and Davis remained convinced that the two foreign governments could be convinced to openly support the Confederate war effort with more than just words. A victory on Northern soil, it was believed, might be the impetus needed to convince the British in particular (the French already gave tacit support and had Britain been willing to intervene openly the French would have followed suit) to come to their aid. Even in the case that it didn't it was hoped that another victory would convince those two powers to further step up their diplomatic pressure on the North and Lincoln further weakening his position in the coming election.
To alleviate pressure on the South, both militarily and economically. The South was struggling greatly and in particular northern Virginia was a wasteland and unable to sustain his army while the North licked their wounds and resupplied comfortably in Maryland and Pennsylvania. Thus Lee was faced with a dilemma between withdrawing further South closer to Richmond or into the Shenandoah Valley which would leave Richmond exposed to a quick thrust by Northern forces. His other option was to remain in the wasteland of northern Virginia and pray that the quartermaster's department would spare enough to sustain his army. Neither option appealed to Lee, the former would hurt Southern planters who hadn't had a crop harvest free of Northern incursion in 3 years and force Lee to withdraw following a win something he was loathe to due. The latter required trust he didn't have in the quartermaster's department. Thus, Lee divined a third option, invading the North, which coincided nicely with Davis belief in a need to alleviate pressure on Vicksburg and Chattanooga. Davis believed a show of force might scare Lincoln into recalling thousands of troops from out West to contain the wily grey fox that was Lee and Lee was only too happy to oblige
Lee wanted a way to resupply and feed his army as well as generate new recruits. Outside of a brief incursion into Maryland that ended at Sharpsburg, or Antietam if you prefer, the North had been untouched by war. Excellent farmland in Pennsylvania meant his men could be fat and happy at the expense of his opponent and the various supply depots and armories at places like Harper's Ferry meant that he could resupply his army at the expense of the North, an idea that greatly appealed to Lee. Along with this is the idea that Maryland, which was technically a slave state and sympathetic to the Southern cause, would provide a new pool for recruits and this would bolster his force even more while providing uncertainty in Washington about the loyalty of their subjects right next to the capital's doorstep.
Finally, and most controversially as well as debatable, Lee had begun to believe in the invincibility of his army. Lee admits to this fact, he had begun to doubt the quality of opponent he faced and believed that his soldiers, who admittedly had never lost under his command, would always beat those "boys in blue" regardless of circumstances. This caused him to make several questionable judgement calls in the course of his invasion and during the decisive battle. To be clear this is not a criticism of Lee, it would be easy to say he was wrong after the fact with those blinders, but it would be wrong to say that his overconfidence and hubris played no role in his decisions leading up to and during the actual engagement.
Lee's endgame was simple, force the Union army to attack him and then defeat them as he had always done. It remains difficult to know whether Lee would have crossed the Susquehanna as he never got that far. However, if the Northern army had refused to give battle it is likely Lee would have created as much havoc and consternation in the North as possible before returning South. In any case it is highly dubious that the Northern government could have stood the shame backlash of withholding the army from a fight and thus Lee was likely never in doubt that he would be able to draw them out into a fight and he was just as certain that he would beat them and end the war before August.