You're probably running too much and experiencing an overuse injurie known as achilles tendinitis.

You need to rest and significantly decrease the distance you're running per week. Do not stretch anything that hurts until it has recovered; stretching will further damage the soft tissues. Recovery could easily take a week or more, depending on how damaged it is.

This is one of the most common running injurie. The first sign of it is typically waking up with a some very stiff achilles, but if you're experiencing it even when running, then you're way past that point and your really need to slow down before you destroy them any further.

The reason you don't always feel pain in the achilles is because tendons (unlike muscles) don't actually produce pain when damaged, but instead they just cause inflammation, which only sometimes results in pain.

At first you only have micro-fractures, but eventually this will domino all the way to a macro-fracture, at which point you're probably talking about months of recovery and more.