Even though the law had no power to condemn him to death—he was "dead" to it, as verse 19 says—Paul was still quite active! His life continued to contain a great deal of activity. When Christ was crucified, He then also became "dead to the law" in the sense that it held absolutely no power over Him. The law's power, its threat to a human being, lies in being able to condemn him to death. But once a person has died, as Paul shows in Romans 7, the law no longer has any power over him.
The phrase "I am crucified" shows that it is a continual thing, an ongoing process. Through Christ's intercession, the law's condemning power is held at bay. This is what Christ does in His role as our High Priest. But Paul then clarifies it by saying that he still is very much alive and kicking—he does not simply roll over and relegate all responsibility to God.
He then shows another facet: Once we have made the covenant with God, we have signed ourselves over to Him, and suddenly our lives are not our own anymore. We still have to go through this life, but it is Christ living His life in us that makes us alive spiritually.
Jesus Christ gave up His physical life so that, through His sacrifice, we could be brought into alignment with God; this is what is called "justification." We respond by yielding to the direction that He now gives to us. We are to have faith in Him and the entire process that He is bringing us through—not merely that we are forgiven for our sins, but also that we will be brought to the "measure of the stature of the fulness of Christ" (Ephesians 4:13). It would not be faith if we were resisting Him throughout this life by disobeying Him! It would be a contest of wills: Can God still save me even though I am rebelling? Trusting in God to bring us to a state of completion while refusing to obey Him are mutually exclusive. God simply will not allow someone into His Kingdom who will not willingly submit to living the way God requires (Matthew 3:2; 4:17; 5:10; 5:19-20; 6:33; 7:21; 13:41, 47-50; Acts 28:23; Romans 14:17; I Corinthians 6:9-10; Galatians 5:19-21; Ephesians 5:5; II Thessalonians 1:5; II Timothy 4:1; Hebrews 1:8; Psalm 119:172)!