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665 B
the whole world
This is an exaggeration referring to the world they knew, which in this context is the Roman Empire.
For God is my witness
Paul wants to add emphasis that he earnestly prays for them and God has seen him praying. The word "for" is often left untranslated.
I make mention of you
"I talk to God about you"
always in my prayers requesting that...I may be successful...in coming to you
"Every time I pray, I ask God that...I may succeed...in coming to visit you"
by any means
"in whatever way God allows"
at last
"eventually" or "finally" or "in the end"
by the will of God
"because God desires it"