Get onto the little extension as shown in the image (red jump), then let yourself slide down a little before jumping left and double jumping up (blue jump). Don't worry, there's plenty of room for error as long as you don't rush the midair jump.
Let yourself fall a little bit after hitting your head on the ceiling before making the midair jump, and let yourself continue going left a little afterward before going right. If you try making the double jump too soon, or if you slam the right arrow key as soon as you make the double jump, you'll never hit the ledge in the right way to start climbing.