I don't agree with waiting until you don't have pain during the first few steps in the morning; if I waited for that, I'd never run, but I've been running every day for years (and did 6 marathons last year) with chronic Achillies Tendonosis, as long as I do those eccentric heels drops every day, and wear shoes with a minimum 9 mm heel drop.
Back when I had to take a month or so off because of my Achillies, when I thought it was safe to start running again, I would do the walk/run thing. The Achillies area is notorious for poor blood flow, so after I would get warmed up, it would hurt LESS (not at all). If it kept hurting after 15-20 minutes, it meant I wasn't ready to start running again.
I should also mention that sometimes if I make my Achillies worse by doing too many hills etc and it's quite painful, I sleep in a night splint made especially for Achillies Tendonitis, (it forces your toes to point just slightly upward, to stretch your calf and heel) and it's amazing how well that thing works. I only paid maybe $60 for it; worth every penny.