Was Paul the “chief of sinners” (1:15)? He was not, and neither did he say at the time he wrote that he was.
To start, think about the terminology. “Chief” of sinners. To be the “chief” is to be the head, the leader, the foremost. At the time in which Paul wrote, was he a man that conveyed anything like being at the head of the class in the involvement of sin? Did he give this sense in any of his writings? He did not. If he was not at the head, what are we to make of the terminology? Before we answer that, consider the word “sinner” and what it means. To be called a “sinner” is to live a life of sin. Was Paul, at the time of his letter to Timothy, living a life of sin? There is no reason to think he was, and the onus is on those who think to the contrary.
Paul was not saying that he was at the head of the class of those living in sin, but he was saying that when the Lord saved him, he considered himself to be at the head of the class. The Lord saved him from the darkness that he realized not that he was in, and when he did come to realize it, he submitted to the Lord Jesus in His death, burial, and resurrection (cf. Rom. 6:3-7).
What about when Paul said he was the least of the apostles (1 Cor. 15:9) and least of the saints (Eph. 3:8). Two things to remember: first, there is a difference in measuring yourself to be the least in comparison with the most (chief); second, in both references, Paul did not say he was guilty of living a life of sin, only that he humbly recognized that in comparison with the other apostles, he was a man untimely born and full of gratitude the Lord recognized in him a reliable servant.
Do Christians sin? They do sin, but not one of them should be called a sinner, for then one would be living the life of sin, and that is not the life the Lord called each to live.