Hi Anton,
seems like we both have the same issue.
The inflammation of the biceps tendon is in most cases due to an Impingement Sydrome in the shoulder.
Impingement Syndrome is responsible for round about 90% of all shoulder issues.
There is too less room for the tendon in the supraspinatus channel to move freely if the arm is lifted overhead.
So movements like pull-ups, pushups, dips and similar movements are very agressive for the shoulder.
You should immediately avoid everyting that hurts your shoulder/tendon to stop the Inflammation.
You can cool the tendon with icepacks and you can take antiinflammatory drugs like Ibuflam or Diclofenac.
Once the inflammation has stopped you should strengthen your external rotator cuff muscles.
You can do this with a Theraband and you should be supervised by a physical therapist.
I for myself am exercising with a tool called "Schulterhilfe" http://www.schulterhilfe.de/produkte/schulter-selbsttherapie-sbt-schulterhilfe/
Maybe you can find something similar in your area.
The exercise with the Schulterhilfe is similar to an exercise called "Cuban Press".
However I strongly recommend to stay away from cuban presses as these worsen the problem due to the lack of support for the upper arm.
This support is very important to isolate the external rotators.
I hope that this is helpful for you.
Best wishes
Markus