What are signs that you have an Achilles tendon tear?

If Achilles is torn. It is very difficult, or impossible to lift your body up on the affected side while standing on that foot only. To clarify: while holding on to the arm of a chair pick your good leg off the ground. Try to do a toe raise on the bad leg. This is usually not possible with a torn achilles.
Achilles Tear. Acute onset of pain upon landing from a jump or pushing off to walk or run. Partial tears may be painful with difficulty walking. Complete tears make you unable to pushoff normally. And MRI or musculoskeletal ultrasound can be helpful to determine the grade of a partial tear or to see if it is completely torn.
Try this. You need a doctor and imaging to verify the diagnosis, but the thompson test is a physical exam maneuver that may help determine it. Google thompson test video and check the image to the left. Go see a doctor!
Inability to pushoff. Complete achilles tendon ruptures prevent the patient from pushing off on the affected limb. Acute injuries typically have swelling and a palpable defect at the rupture. Often times they are not painful since there is no tension on the complete tear. Partial tear are more painful.
Pain. Usually you would have pain. Best way to detect severity is MRI and if you can feel a gap or dell in achilles.
MRI. The most definitive way to tell would be rupture detected by an mri. If you are unable to plantarflex and dorsiflex your ankle, that would be the main clinical sign.