icing and rest are fine but what cures the achilles tendonitis is eccentric calf raises. you stand half on a step, facing up, raise yourselves up to your toes, then lower yourself very slowly (the calf goes up as you lower yourself) with the bad leg until your heel is down below the step and fully extended, and repeat. 3 sets of 10 reps per day will be enough to get you healthy within a week or two, while still running but not pushing it a great deal if the heel is really feeling problematic. You can add weight while doing them. There are leaper machines that do this: you stand on a bar, with your shoulders under another bar with the machine set to maybe 30 to 150 lbs, depending on the strength of your calves, and you do the same thing.
Strengthen your calves too quickly and you will get shin splints, so every day add the opposite exercises: on a step facing down you leave your heel on the step while rotating your foot up and down below the step, many, many times per day.
Honestly, I do not think that some of the elites or their trainers know that this is the only treatment for achilles.