# not moved away from the hope of the gospel that you heard

"continuing in the hope of the gospel that you heard" or "continuing to confidently expect what is promised in the gospel". 

# which was proclaimed

"which believers proclaimed"

# to every person created under heaven

"to every person in the world". God created everyone.  

# the gospel of which I, Paul, became a servant

"the gospel that I, Paul, serve God by proclaiming". Paul was a servant of God.