# asked him for a favor
Here the word "him" refers to Festus.
# that Festus might summon Paul to Jerusalem
This means that Festus would order his soldiers to bring Paul to Jerusalem. Alternate translation: "that he might order his soldiers to bring Paul to Jerusalem"
# so that they could kill him along the way
They were going to ambush Paul, but Festus was not aware of this part of their plan.