1 Chronicles 11:41 lists Uriah the Hittite as one of David's mighty men. Would this indicate a relationship with David prior to the incident with Uriah's wife? How well would they have known each other? If so they would've known each other prior to that incident, how would this change our interpretation of the events surrounding that incident?
Uriah and David did know each other, as you suggest. It gives us insight into David's struggles and failures. It is hugely encouraging that God isn't surprised by our failures but loves us none-the-less.
2 Sam 11 tells us the tale. David, rather than leading his army as was the custom, sends them to battle and he stays in Jerusalem. He saw the woman washing herself (after her menstrual cycle presumably) on her rooftop. This of course raises the question of whether Bathsheba was not in fact advertising herself to a king whose balcony overlooked her rooftop. She also seems to come quite willingly to his request when he sends after her and sleeps with David. Of course the blame still falls on David but it is unlikely Bathsheba was innocent. We find that she later pulls strings to make sure her children go on to success. She becomes pregnant with David's child while her husband was gone, which was punishable by death. He sends for Uriah who was a favored officer of David. David fakes the meeting as a way to learn of the battle and treats Uriah really well, trying to get Uriah to go sleep with his wife so they child can be attributed to Uriah. Uriah refuses and does not go home. David tries to get him drunk and Uriah still wont go home.
The point we get from this is that Uriah and David were well known to each other and that Uriah was fiercely loyal to Israel and David, despite being a Hittite. David then feels he has no alternative but to destroy Uriah and marry Bathsheba to remove the public knowledge of adultery. He instructs Joab to cause Uriah to go to the highest risk point in the battle. Uriah died in the fight.
Joab even tests David's intention based on how he responds to Uriah dying, again indicating their personal relationship, and unfortunately, David doesn't seem upset at all and is glad at the news.
Bathsheba then marries David and the child is born. She then has 3 more children, including Nathan and Solomon (both of which are in the lineage of Christ, one genetically (Luke) and one royally (Matthew)).
Again, regarding knowing each other previously. David was fully aware of who Bathsheba was and had prior history with Bathsheba's family members. She was the daughter of Eliam and married Uriah. David kills Uriah and takes Bathsheba as another wife. Absalom (David's other son by another wife) leads a rebellion against David under the advisement of Ahithophel, who was initially a counselor to David. Ahithophel is the father of Eliam, or the grandfather of Bathsheba. Ahithophel has never forgiven David for defiling Bathsheba and murdering Uriah.
So long story short, Yes, they did know each other previously, yet David didn't care and destroyed that relationship to satisfy his lust and as a result, his entire family was destroyed over and over, with rare single exceptions given to preserve the line. The sword never departed from his house because of it.
David’s Mighty Men
1 Chronicles 11:10
“Now these are the chiefs of David's mighty men, who gave him strong support in his kingdom, together with all Israel, to make him king, according to the word of the Lord concerning Israel. “
The following verses make it clear who David’s inner circle included; his mighty men.
1 Chronicles 11:41 “Uriah the Hittite, Zabad the son of Ahlai.”
This verse, in particular, reveals Uriah to be one of David’s mighty men.
It is unlikely that David was ignorant of this fact; he had, himself, appointed them.