This is a great, quite safe route that runs almost entirely along bicycle paths or wide-shouldered roads. Begin in Sea Point at the end of the Greenpoint Stadium/2010WC FanWalk, which runs all along Greenpoint Main Road and has a dedicated bike crossing (across Buitengracht) and down Waterkant St, which this map won't show.
Remaining on the bike route (Waterkant St), you continue to the Adderley St, where the bike route has a ramp up onto the wide pedestrian crossing over Adderley St to Cape Town Station.
Cross this, but turn into Adderley Street towards the mountain, then immediately left onto Strand St. Strand St will take you along for a distance and you will face a fork - one branch is a highway onramp, to the left; to the right, continue straight and level under the elevated highway until you join up with Albert Road.
Follow Albert Road for a long time until you find the a road called Malta Road veers off to the left and becomes a bridge over the railway line. Take Malta Road as it slowly curves around. Very soon, Malta Road becomes Liesbeek Parkway and gains a safe and comfortable bike route to its immediate left.
This will take you all the way to Campground Road via the old, 1970s Rosebank-Newlands bike path system.