I have pain only when my Achilles tendon is relaxed, what could this be?

You have tendonitis. The most common cause is sports injury, the second most common cause in elevation of uric acid , if more than 6.0. The third common cause low thyroid, check tsh, if more than 2.0 is abnormal. In all you need treatment. Can try ice and strech.
Achilles tendinitis. When the achilles tendon is relaxed, such as when sitting, or waking up in the morning, the blood flow is lessened, resulting in pain. As you exercise the area by walking, or participating in more strenuous activities, the blood flow increases, warms the area, and, makes it feel better. You may need physical therapy to resolve the issue.