I don't have a good map handy, so this is from memory. Check a paper map before you go.
Follow Orcas out of the park (or, for a less steep hill, take a right after one block of steep uphill, I think onto Dawson, then after you reach the crest, make your way back to Orcas), and up to the top of Beacon Hill. Just before Beacon, turn right onto the Chief Sealth Trail. That will take you to 15th Ave South, which is more busy than I like, but has never given me trouble or felt too unsafe. Take that north (or go a block over and enjoy less traffic). Take the 15th Ave Bridge, then go left into the ID, and make your way to the waterfront. Take the lower bridge, go right, and follow the trail it to Alki. Sorry I don't remember the road names here, it's been too long.
But it's a really nice ride.