David, in convincing Saul why he should be allowed to be Israel’s representative on the battlefield says,
“Your servant has struck down both lions and bears, and this uncircumcised Philistine shall be like one of them, for he has defied the armies of the living God.”
And to Goliath he says,
You come to me with a sword and with a spear and with a javelin, but I come to you in the name of Yahweh of hosts, the God of the armies of Israel, whom you have defied. This day Yahweh will deliver you into my hand, and I will strike you down and cut off your head. And I will give the dead bodies of the host of the Philistines this day to the birds of the air and to the wild beasts of the earth, that all the earth may know that there is a God in Israel, and that all this assembly may know that Yahweh saves not with sword and spear. For the battle is Yahweh’s, and he will give you into our hand.”
So, in both of these, David gives his reason for wanting to fight as an opportunity for God to show His hand, to have victory or vindication over the gods of the Philistines. In the next chapter, David doesn’t demand the promised reward. Even when the promised daughter is given to another man, he doesn’t push the matter.
And David said to Saul, “Who am I, and who are my relatives, my father’s clan in Israel, that I should be son-in-law to the king?” But at the time when Merab, Saul’s daughter, should have been given to David, she was given to Adriel the Meholathite for a wife.
Later, Saul offers him another daughter and still David waits. He recognizes the weight of being the king’s son-in-law and doesn’t just pounce on his opportunity to “climb the ranks”. Also, throughout the rest of Saul’s reign David has several chances to take the throne, but waits. His actions support the stated motive.