The Tasman Peninsula has an absolute abundance of dolerite. Although a lot of it is choss, there is plenty that is good quality. From the amazing sea stacks of the Totem Pole and Candle Stick to the modern sport climbing of Mt Brown, the area has cliffs everywhere. There is huge potential for as much climbing as anyone could want, the only drawback are the long approach walks to the undeveloped areas. Despite this, the development of many new sport climbs at Mt Brown and the climbing of the Free Route on the Totem Pole has seen a resurgence in climbing on the Peninsula, as has the recent development at Cape Raoul. Cape Pillar and Tasman Island have had quite a few routes put up over the years, but details are sketchy.
Port Arthur is a bit over an hours drive from Hobart. The Candle Stick and the Totem Pole can be reached by an hour and a bit's walk to Cape Hauy from Fortescue Bay, which you turn off to a couple of kms before Port Arthur. Mt Brown and Dauntless Point are half an hours walk south from the Remarkable Cave car park, just south of Port Arthur. After walking along the track for 25 minutes you come to a sandy creek. From here follow the coast around, first passing Parrot Shelf Cliffs, and then coming to an amazingly steep and juggy (for dolerite) cliff covered with eye bolts - this is the Paradiso.
Long and short dolerite sea cliffs. The rock is quite different from alpine dolerite because it has been weathered by the waves, not snow and ice.
There is good but expensive camping at Fortescue Bay and you could also possibly camp on the flat areas on the way to Mt. Brown (but there is no water). The area has recently been declared a national park so entry fees now apply.
Most of the climbing on the Peninsula depends on swell size more than anything. Generally the swell is bigger in winter, but there is nothing south between the Totem Pole and Antarctica, so big swells are possible (and even likely) all times of the year. The Totem Pole is a deep, shady, chasm and only gets sunlight for a couple of hours a day, so the summer months are best for an attempt (December to March).
There are some great coastal walks on the Peninsula that take in some spectacular scenery. Cape Raoul and Cape Pillar are great, and the walk out to Cape Hauy to contemplate the singularity of the Totem Pole is a must even if you don't plan on attempting the thing. There are also some funky caves and other natural features that don't require walking, including Tasman Arch, Remarkable Cave, and the Tessalated Pavement at Eaglehawk Neck. The Port Arthur historic site is a grim reminder of Tasmania's history as the penal colony of Van Dieman's Land and is worth a look.
Dauntless (15), The Candlestick (16), Inferno (17), Aquaphobia (22), Pole Dancer (22), I've Heard It All Before (23), Talk is Cheap (24), Offender of the Faith (24), The Totem Pole (25), Déjà Vu (28).