Remove the lockset from the door by undoing the two retaining screws on the inside knob. Pull the two halves of the lockset out of the door. Set the inner knob aside. The outer knob containing the lock cylinder has a long round piece projecting from it. This is the spindle. The spindle consists of two parts. One part is a half-round cylinder that works the latch. The other part is a round tube that connects the inside lock button to the outside locking mechanism. The round part is what you must remove to change the lock. The half-round part is not removable.
Remove the key from the lock before removing the spindle. The spindle is held in place by a metal tab, retaining plate and spring catch. Use the pointed end of the removal tool to align the tab so it is perpendicular to the bottom of the half-round portion. Then insert the pointed end of the tool into the end of the round part to turn the entire spindle assembly until the tab lines up with the two round stems where the retaining screws go.
To release the round spindle, insert the pointed end of the removal tool at an angle against the stem that’s beneath the half-round spindle portion and slide it under the edge of the retaining plate. Apply pressure inward toward the spindle to release the spring catch. Keep the pressure on and tilt the spindle down. The round part should slide right out. If it sticks, keep pressing with the tool and use your thumb to slide the spindle out. If the round part rotates before you remove it, start over and realign the spindle.
With the round spindle removed, you can remove the lock cylinder. Place the forked end of the removal tool perpendicular to the half-round spindle and slip the upper fork into a hole at the top of the retaining plate and pry open the spring under the plate. With the spring open, slide the tool all the way into the opening until it stops. Put your other hand over the outside of the knob so it covers the lock cylinder. Push on the removal tool and the cylinder should pop out in your hand.
When you are done rekeying the lock cylinder, remove the key, align the pin tumbler cover with the channel at the top of the knob cavity and slide the cylinder back in until you hear the retaining spring click into place. Reinstall the round spindle by sliding it tab-end first and tab up into the half-round spindle. Use the pointed end of the removal tool to push the round spindle in until you hear a click. Reinstall the lockset in the door.