〜てく means “…and go” / “take along” / “keep on”. The い of 〜ていく drops out: 持っていく becomes 持ってく, 出ていけ becomes 出てけ, 帰っていった becomes 帰ってった. The same い-drop as in 〜ている → 〜てる.
Is 〜てく rude? Who can you say it to?
With friends: OK. With coworkers: careful. With a boss or teacher: avoid. Neutral casual. Used all the time in everyday talk.
How common is 〜てく in real speech?
In 1,905 live comedy routines, 〜てく appears in 88 lines from 62 comedians.