go to, follow
趁 centers on moving toward something or someone: going to a place or following after a person or path.
趁 combines 走 (run, move) with 㐱, which likely contributes the sound and may relate to following or pursuing. The exact historical development of 㐱 is uncertain.
The running enclosure 走 suggests movement, and the inner 㐱 looks like a person with a distinctive shape. Picture someone running to catch up with that person: going to them and following closely.
For チン, imagine a chin-up bar at the destination: you go to the bar and follow a workout routine. Chin -> チン, and the running motion fits the kanji's meaning.