Kirksey was taken with the No. 71 overall pick on Friday night.
The 6-foot-1, 233-pound Kirksey is expected to play inside linebacker for Cleveland. He had 105 tackles last season, playing outside linebacker for the Hawkeyes. He made 38 starts and played in 49 games in college.
Earlier, the Browns selected Nevada offensive tackle Joel Bitonio at No. 35.
The Browns lack receiver depth and there is a report that Pro Bowler Josh Gordon is facing a possible suspension for failing a drug test. There were seven wide receivers taken in the second round.
Cleveland had four picks remaining.