1 Samuel 31:4-5
Then Saul said to his armor bearer, "Draw your sword and pierce me through with it, otherwise these uncircumcised will come and pierce me through and make sport of me." But his armor bearer would not, for he was greatly afraid. So Saul took his sword and fell on it. When his armor bearer saw that Saul was dead, he also fell on his sword and died with him.
2 Samuel 1:8-10
"He said to me, 'Who are you?' And I answered him, 'I am an Amalekite.' Then he said to me, 'Please stand beside me and kill me, for agony has seized me because my life still lingers in me.' So I stood beside him and killed him, because I knew that he could not live after he had fallen. And I took the crown which was on his head and the bracelet which was on his arm, and I have brought them here to my lord."
There is no logical contradiction here.
While exactly what happened is not perfectly clear, there are at least two possibilities.
It is possible that the armor bearer fled after Saul asked him for a mercy killing, then Saul fell upon his own sword, as we see in 1 Samuel, and then the Amalekite found him not yet dead though mortally wounded, and killed him as he asked. The dead body was then later discovered by the returning armor bearer, who was so distraught that he killed himself.
It is also possible that the Amalekite found Saul dead by his own hand, stole his crown and bracelet, then went to David, the enemy of Saul, and lied about having killed the former king, expecting some kind of reward. Interestingly enough, David did not want Saul killed as God had anointed Saul the king of Israel, and so he executed the Amalekite for murdering the king on the basis of his own testimony, rather than rewarding him.
If one passage stated authoritatively that Saul killed himself, and another stated authoritatively that it is not the case that Saul killed himself, then there would be a contradiction. Neither passage authoritatively states that Saul killed himself, and neither passage authoritatively states that it is not the case that Saul killed himself.
1 Samuel states that Saul fell on his own sword, and that his armor bearer saw Saul dead. We don't know what may have happened between these two events. While the natural reading of the narrative is that the armor bearer was present when Saul fell on his sword and remained present until he died, that is really only inferred from the text, not explicitly stated within it, and therefore cannot be taken as certain.
2 Samuel states that an Amalekite claimed to have killed Saul. We have absolutely no idea if he is telling the truth or not.
So there is nothing even remotely like a contradiction here.
Indeed, the Christian worldview is the only worldview that is not inherently contradictory.