For diving, best is just before the fall storm season. Kelp is still alive and attached and vis can be up to 100 feet. Next to last time I was at Browning wall you could look up and see the trees on the bank from 100 feet. This was in November and a couple of days later we left in a storm with heavy winds and rain.
However for an all round visit I would pick mid to late September. The diving is very good and the weather for outside activities is also very good. Starting to cool off, but still late summer conditions. A month before is good and a month after as well but you run the minor risk of a late algae bloom in early August or an early winter in late October.
For just diving October - November is slightly better, but hiking and biking will be a bit riskier as sometimes we get an early winter which means the rain starts early.
Sometime in November storms start to come in and some of the dive spots are not really available. Have been blown out of a trip to Browning Pass because of hurricane force winds that did not allow the boat to get there. Diving is still available, but options become more limited when the wind is blowing. Hiking and biking is also marginal in storms.
Not sure what level of hiking/biking you are looking for but here are a couple of suggestions.
A very easy 55km bike ride that is very scenic in the Victoria area can be found here
Galloping Goose Regional Trail The benefit of this ride is that is an old rail line so very little traffic and "fairly" level.
Options for hiking on Vancouver Island range from day hikes all up and down the island to the West Coast trail a six day trail for experienced backpackers that requires a permit. Spectacular trail. Here is some information.
west coast trail preparation guidebook map
There is a ski hill on Vancouver Island where you can go biking/hiking in the mountians in summer.
BC Summer Activities | Discover Mount Washington It is easy to dive one day and hike the next. In winter a dive and ski day is easy.
Whistler/Blackcomb is more world class and if I was just going hiking and biking I would probably go there. For skiing Whistler is second to none. However this is not on Vancouver Island and the diving on the Vancouver side is not as nice. Most of Vancouver is on a river delta so diving is more limited. There are a couple of sites but Vancouver Island is significantly better.
Feel free to PM me if you want more detailed options.