Was David guilty of adultery (with Bathsheba) and murder (of her husband Uriah)?
The Talmud (Shab. 56a) explains that David’s marriage to Bathsheba involved neither adultery nor murder. The soldiers in David’s army gave conditional divorces to their wives before going to war, so if they were missing in action their wives would be able to remarry. Since Uriah in fact didn’t survive the war, his divorce was retroactively effective, so Bathsheba was a divorcee.
David’s sending Uriah to the front certainly wasn’t murder; he was killed by the enemy. The Talmud says that David was justified in telling Joab to arrange for Uriah to be killed in battle, because Uriah had acted disrespectfully when he referred to "my lord Joab" while speaking to David (2 Sam. 11:11). But we see that Nathan the prophet rebuked David for causing Uriah to be killed (2 Sam.12:9), so it seems that David wasn’t entirely justified in doing so; see also 1 Kings 15:5.