Captain Miller showed personal courage by just getting off the boat in the beginning of the movie. Personal courage is facing fear, danger, or adversity (physical or moral). Seeing all those soldiers being shot down around him, having an explosion really close to him and still being able to lead his troops shows personal courage.
Captain Miller also showed personal courage when he lead the way all the way, up to where the enemy was. Even though the enemy was firing nearby, he still faced danger and more than likely faced fear to get to where he wanted to be.
One thing I did notice was after the sniper took out the enemy in the tower; Captain Miller ordered his troops to go into the trenches. He ordered the men into the trenches but he stayed behind. I am not sure if maybe I missed something, but if I am correct, then he was supposed to go with the lead element.
If I am right, then he did not do his duty or fulfill one of his obligations. Captain Miller did however redeem himself by standing up and making himself a decoy so that Private Jackson could advance without drawing fire. In doing that, he also showed personal courage.
Captain Miller showed loyalty to the Army when he reminded the troops or their duty. A bunch of them were commenting on how they thought that the mission was "foobar" and Captain Miller just reminded them of their duty and gave a little explanation about it. In addition, when he told the troops about how complaints go up and never down. I think that that can also be considered his duty as well as him showing loyalty. He showed loyalty by not bashing the army in front of his subordinates.