in the line of duty

[Theseus]

As a consequence of all that is normally required in some area of responsibility:-
I'm sorry for your loss. You will be proud to know that he died in the line of duty.
στη γραμμή του καθήκοντος; κατά την εκτέλεση του καθήκοντος;

 

[Palavra]

Your first version (*στη γραμμή του καθήκοντος) is a calque that is not used in Greek. The second one is used, however, along with πέθανε/έπεσε στο καθήκον.

 

[bernardina]

You could even go one step further and write θυσιάστηκε στο βωμό του καθήκοντος which is, to me, a little more touching

 

[dharvatis]

Και εν ώρα καθήκοντος νομίζω;

 

[Zazula]

The formal expression (found in Laws etc.) is κατά την εκτέλεση διατεταγμένης υπηρεσίας (και ένεκα αυτής).