don, arrive, wear, counter for suits of clothing
着 centers on the idea of something making contact and staying in place: putting on clothes, arriving at a destination, or attaching to something. The counter for suits of clothing is a natural extension of the 'wear' sense.
arrival; arriving at ...
arrival
to arrive at; to reach
bathing suit; swimsuit; swimmers
underwear; undergarment; underclothes; lingerie
wearing (e.g. a uniform, seat belt, helmet); having on
to wear (from the shoulders down); to put on
sticking (in one place, position, etc.); settling; fixing; adherence
equipping; installing; fitting; mounting; putting on
changing clothes; change of clothes
newly arrived; newly received
to calm down; to compose oneself; to regain presence of mind; to relax
to put clothes on (someone); to dress; to clothe
landing; alighting; touch down
conclusion; decision; end; settlement
close adhesion; sticking firmly (to); being glued (to)
calmness; composure; presence of mind
kimono; Japanese traditional clothing (esp. full-length)
starting work (on); setting to work (on); setting about (doing); beginning; commencing; embarking on
coat; jacket; outerwear
attachment; adhesion; insistence; tenacity; fixation; obsession; clinging (e.g. to old customs)
cohesion; adhesion
arrival and departure
landing (on the ground)
着 is the modern simplified form of 著. The older form combines 艹 (grass) with 者, which likely contributed the sound. The modern form replaces 艹 with ⺶ (a variant of 羊) and retains 目, but the historical connection to 著 is clear.
The outer ⺶ looks like a sheep, and the inner 目 is an eye. Imagine a sheep's eye watching you as you arrive and put on your coat—once it sees you, you're attached to the spot.
For チャク, picture a chick arriving at a party wearing a tiny suit: chick -> チャク, and the chick is dressed to impress.