Install the bamboo flooring up to the plank or row before the plank that will go around the door frame.
Cut the door casing at a height of 9/16-inch or at a height where the bamboo floor can slide easily underneath it without a visible gap between the casing and the floor.
Measure the corner of the door frame and mark the back of the plank where you will need to cut. Make sure you factor the appropriate expansion gap into your marks -- typically between 1/2- to 3/4-inch. Measure twice, cut once.
Cut the plank at the mark with a jigsaw. Use a table or miter saw to rip or make larger cuts in the plank if needed.
Fit the plank around the frame, before attaching it, to make sure it fits properly.