I used to commute from Venice to Korea town all the time. It's a breeze. Take Venice Blvd with it's very wide bike lane from Venice to Cochran. From Cochran, take a right on 8th street and then a left on Detroit Street. Detroit is one block west of La Brea. Take Detroit North til you get to Melrose area. I know that route til 4th street and Cochran and Detroit cross the big streets with traffic lights along that route.
For a longer more traffic free route, you can take beatiful and scenic Bollana Creek bike path from Marina del Rey til it ends on Overland (I think). From the end of the path, go right/North past Washington to Venice.