Good question, since there is no direct bike path between the two, and you essentially have the bulk of downtown in-between, so there's no way to avoid cars.
If say, you work north from the canal, a good strategy would be to intercept the de Maisonneuve path (runs east-west), and head east until about Jeanne-Mance. Head north and once you cross Sherbrooke, you can jump on Parc which is one block west of Jeanne-Mance. Continue north on Parc, and you'll bump into short section of path that will bring you to the main park entrance.
If you enter Mont Royal from there, you can climb the gentle slope of Olmstead up into the park. Note that this is unpaved (hardpack / gravel) and the pedestrian traffic is thick. But you'll have easy access to the main lookout, and you can pass near the cross close to the summit.
If you continue north and turn west at Mont Royal (the street), you can climb the road that crosses over the park - Camillien Houde. This is a steeper climb, and you'll have a bit of road shoulder to use, but the shoulder disappears as you approach the peak. Car traffic is not super thick on this road, so you should be OK if you don't mind co-mingling with cars. After you pass the peak, you'll descend a bit until you see parking lots to the left, where you can enter the park.