1 Samuel 16:22. "And Saul sent to Jesse, saying 'Let David remain in my service, for he has found favor in my sight.'" David's favor with Saul will not last; Saul will instead develop a fierce feeling of rivalry with David, the result of which will ultimately be David taking the throne of Israel from Saul. It's important to note, though, that this was not because David was seeking to take the throne, but because it was given to him by God, who saw in David, even as a child, a king who would serve Him and His people.

David never sought the kingship, he only sought to serve his king, both Saul as his earthly king, and more importantly God as his ultimate King. But David had been anointed by Samuel, at God's direction, to be Saul's replacement. Saul did not know this when he asked David's father to let David stay on in his service. David obviously knew he had been anointed, but he did not know how it was all going to play out. Maybe David assumed God was placing him in Saul's service to prepare him to be Saul's successor in a peaceful way. Neither Saul nor David, at this point, saw what was coming, but each had the choice to act in service of the King. 

None of us know how God's plan is playing out in our lives (although often we think we do!). All we can really do is remain in His service - to be obedient to God and His rules in every instance of our lives. Saul strayed from that obedience, and lost the throne and his life. David strove to be obedient (even to the point of not killing Saul, his king, when he had the chance), so David gained the throne and a mighty kingdom. Even during his reign, however, the kingdom was reliant on David's proper service and obedience to God. Because of David's relationship with God and his spirit of service, Israel became a mighty nation under his rule.