I commuted this route for 3 years.
When Southern are running normally, it's a 14 minute journey into London Bridge and about another 10 on the Jubilee Line to Canary Wharf, it used to take about 35-40 minutes all in.
You're very likely to get a seat on the train from North Dulwich; the Jubilee Line from London Bridge is rammed and unpleasant (but mercifully quick). At London Bridge you should be able to get on the first train.
The Overground is reliable, but you'll never get a seat. It's air-conditioned and pretty tolerable. The interchange at Canada Water is crowded and you usually have to let a few Jubilee Line trains go before you can get on. But then it's just one stop. And avoids zone 1, so is only £1.70 each way PAYG.
The interchange at Shadwell takes a good few minutes, the DLR is just as crowded as the Jubilee Line, and it takes a fraction longer overall.
It's about 20 minute walk from North Dulwich station to Denmark Hill, or the 42 bus, so you should factor that in.
I have cycled every route - via Greenwich foot tunnel is quickest (probably around 35 - 40 mins from North Dulwich station), and the road up from the foot of the Isle of Dogs to Canary Wharf is relatively quiet.
I would take a few sample journeys to find out what works for you. Apart from cycling, I preferred the route from North Dulwich to London Bridge, as it was nice to get a seat and the trains are not too rammed. But since early this year Southern have been utterly undependable (one train an hour this morning instead of every 10 minutes!).
Edited 1 time(s). Last edit was 2016:12:21:09:17:57 by Lowlander.