No.  Paul is speaking of what is lacking in his ministry.  

Obviously, Christ’s sufferings are fully sufficient for salvation; therefore, scripture could not possibly be saying that the church (Christ’s body) needs to do more in order to secure salvation.  The sufferings of Christ that are shared by those who comprise His church are part of what theologians call progressive sanctification.  Progressive sanctification is the work that God does in a believer’s life in preparation for eternity.  Its purpose is to “conform us to the image of His Son, Jesus Christ.”

Since we are already part of “His body,” the conforming work is really a perfecting or completing of that transformation in the physical and temporal realm of creation.  As we become more like Jesus we will experience Him more and more.  This experience inevitably involves suffering, just as scripture warns.  There were a lot of things “lacking” in our lives prior to salvation; namely, righteousness.  With righteousness there comes suffering because someone must shed blood in order to atone for the sin that begets unrighteousness.

Paul is really trying to proclaim that he hasn’t suffered enough and that he willingly accepts this in light of the gain that is found in Christ Jesus.