How can I treat my own Achilles tendon injury?

Several items. As long as you are sure that it is not torn, you can rest it, use ice, compression and elevation. You may want to put a pad in the heel of your shoe to take pressure off the achilles tendon when standing & walking.
Don't. While most doctors do not want to see every person who has an occasional ache or pain which is a normal part of life, an achilles tendon injury is not something to manage yourself. You should find out how severe the injury is first. A severe injury can progress to a rupture of the tendon which then requires surgery. This is why we go to medical school; to help people like you.