Hi Johanna,
I think you would know from the context whether it meant to chase after or to sue someone...
For to sue you could also say -
Poursuivre en justice
Je l’ai poursuivi ( en justice) pour diffamation = I sued him for libel
Or you might use -
Engager des poursuites = to start proceedings
Sometimes poursuivre just mean to continue -
Avant de poursuivre , j’aimerais que ...= Before we carry on, I’d like...
So a very useful verb to learn...