By plane
Air Nugini (http://www.airniugini.com.pg/) has four flights a week from Port Moresby to Buka Island, from where a short water taxi ride ride gets you to the main island of Bougainville. The flight takes about two hours, and like all travel within Papua New Guinea, is expensive.
By ship
Truly adventurous types might want to try securing passage on a weekly ship which departs from Rabaul on New Britain island. Try Star Ships tel: +675 97 9821070. It is unlikely that you would be able to make a formal booking; rather just turn up in Rabaul and have plenty of time on your hands.
There are reports of travellers arriving by boat at Buin from Korovou in the northern Solomon Islands. The immigration situation is complex though, and this route is not encouraged by officials from either nation.