The best LR resource for all things Achilles is from Ciarán Ó Lionáird:
Before I dive in: What follows is for insertional Achilles tendinopathy, not a true injury to the tendon itself. In other words, the tendon is not ruptured or damaged. It just ... hurts. The former requires surgical intervention most of the time. My advice is for the latter kind of problem.
I struggled with the same injury for about a year. The "aha" moment was when I learned that the fibers in the tendon are disordered -- not injured. Obviously this is not a clinical explanation, but ... the tissues get all messed up and are not able to heal properly from load. So they get stiff, send pain signals to your brain, etc.
To fix it you have to reorder the tissue (and, if possible) make everything stronger.
The Smith machine at the gym should be your new best friend. Do eccentrics on that three times a week: two feet up, then down slowly on the injured leg. When I started I could barely do the bar. Four months later I was doing 200+ pounds, and my tissue was strong enough to handle regular running.
My achilles isn't perfect -- it is still stiff as sh*t in the morning and gives me occasional pain. But thanks to all that work it is strong enough to handle 80-mile weeks.
One other thing ... a lot of this is in your head. Your body/brain connection are trained to believe something is wrong in that tissue. You have to train it to fire properly, to work properly. Your body/brain have to relearn how to fire the right way. Which is a long way of explaining why rest doesn't work for sh*t. It isn't retraining anything. You have to progressively load the tissue to fix the tissue.
Finally, there is an Aussie PT with amazing resources on insertional Achilles issues. Great advice on how to do what I describe above (with much better explanations.)