Achilles tendon problems are pretty common among cyclists, as I found out after I developed one and started researching. Achilles tendinopathy is classified by the American Physical Therapy Association as "Self-reported localized pain and perceived stiffness in the Achilles tendon following a period of inactivity (ie, sleep, prolonged sitting), lessens with an acute bout of activity and may increase after the activity".
If this sounds like what you have, then moving your cleats (all the way) back in your shoe will help to take the load off the tendon. Heel raises are the most recommended exercise/treatment (stand on a step with just the balls of your feet and toes, then lower your heels slowly as far as you can to get a good stretch, then raise up slowly as high as you can). Do this 1-2 times a day, 3x10. I did them first thing in the morning.
Anti-inflammatories like ibuprofen help with the pain, but don't appear to help it heal any faster. You can ice your tendon(s) after rides.
It took a good 6 months to resolve for me. I still have occasional minor soreness after long and/or hard rides. But now I can at least walk down the steps in the morning! My soreness developed after I got a new pair of shoes, so cleat placement may have been an issue. And, it also started after I got a single speed and began really grinding up a lot of steep hills. Who knows.
If you have painful thickening of the tendon, you should definitely have it seen. I'm a pharmacist, not a physical therapist or a physician, so use your own judgement.
Good luck!
Ref for guideline: J Orthop Sports Phys Ther. 2010:40(9):A1-A26. doi:10.2519/jospt.2010.0305;