"that conforms to the glorious gospel of the blessed God, which he entrusted to me." NIV translation
Paul completes this list of who the law applies to with the fact that it applies to anyone that opposes the gospel that was given to him. So, we might ask ourselves what gospel is he talking about. Paul is talking about grace and the fact that salvation is a gift of God to all who will accept His Son. The law applies to all who do not accept this gift and instead want to work for their own righteousness.
"I thank Christ Jesus our Lord, who has given me strength, that he considered me faithful, appointing me to his service. Even though I was once a blasphemer and a persecutor and a violent man, I was shown mercy because I acted in ignorance and unbelief. The grace of our Lord was poured out on me abundantly, along with the faith and love that are in Christ Jesus." NIV translation
Paul continues by reminding Timothy that he was chosen by God for the ministry not because of his knowledge of the law but because of God's mercy and grace. Paul was a good example of humility as he never forgot where he had been and how God had brought him to where he was in the ministry. Today, there are far too many in the ministry that have grown up too pious and do not remember that they were miserable sinners that have been saved by grace. Once you forget that, you can not effectively relate to us common folks and you end up with a religious system.
"Here is a trustworthy saying that deserves full acceptance: Christ Jesus came into the world to save sinners - of whom I am the worst. But for that very reason I was shown mercy so that in me, the worst of sinners, Christ Jesus might display his unlimited patience as an example for those who would believe on him and receive eternal life." NIV translation
One of the greatest qualities of Paul was that he never forgot what God had done and where he had come from. Here, he explains to Timothy that the reason that God brought him out of his murderous past was to display God's mercy and patience to even the worst of sinners. We are reminded from this that God tends to use the lowly things (and people) to display His power.