How exactly you have updated to LR 6.1.
Did you install creative cloud app to update ?
And also where exactly it is showing you that trial has expired.
A screenshot will be really helpful.
Open LR and Sign In with the Adobe ID you used to buy LR 6. Re-Enter the serial # you got for LR 6 if so required.
Just for clarification, is the warning you are receiving regarding Lightroom Mobile or Lightroom itself. When Lightroom 6 is first installed you have a 30 day trial of Lightroom Mobile. And you will receive a warning that that mobile feature trial is about to expire.