Air Chathams flies from Auckland, Wellington, and Christchurch. Flights are small (usually Convair 580 turboprops) and heavily booked — reserve months ahead, especially during summer. The airline also runs a cargo-passenger service from Wellington that is cheaper but slower.
There is no regular passenger ferry service. The Chatham Islands are genuinely remote: getting there takes planning, and disruptions from weather are common. Build in a buffer day on either end if you have a fixed connection to catch.