# As the truth of Christ is in me, this

"As surely as you know that I truly know and proclaim the truth about Christ, you can know that what I am about to say is true. This"

# this boasting of mine will not be silenced

"no one will be able to make me stop boasting and stay silent" 

# this boasting of mine

This refers to what Paul spoke about starting in 2 Corinthians 11:7.

# parts of Achaia

"regions of Achaia."