"I thank God, whom I serve, as my ancestors did, with a clear conscience, as night and day I constantly remember you in my prayers." NIV translation
Paul continues his reflections by reminding his "son" in the faith that his present situation (death row) is not only the "will of God" but also it is in service to God. He reminds Timothy that his ministry continues even though he is in prison and that God is using him. We also see that he thanks God but it is not for the fact that he is on death row and still able to serve. Paul thanks God for the time that he had with Timothy and probably the fact that his ministry would continue through his efforts.
He reflects on the fact that his ancestors (Jews) had also served God even though they had rejected Jesus Christ. This is a reminder that it is not our actions (service) that brings the gift of eternal life. He goes on to say that he has a clear conscience but, what does that mean? To understand what Paul was talking about, we must remember that he was a Jew of Jews.
In Judaism, the concept of conscience speaks of the nature or essence of our being and the fact that we were created in the image of God. A part of this image (nature) is the inner voice that helps us to discern right from wrong and it is referred to as "yetzer tov". The other part of our nature is called "yetzer ra" and it speaks of our selfish side (our ways). These dual voices acting in our lives cause us to make choices and this is what is commonly called "free will".
A "clear conscience" speaks of the fact that Paul was content with the choice that he had made to put his trust in Jesus Christ. His choice meant turning his back on his religion and embracing the gift of eternal life through grace. Now, looking back, he basically says that he does not regret his choice even though it probably cost him a lot of his family ties.