The Isshin Ashina trophy in Sekiro: Shadows Die Twice can be unlocked as part of the Shura ending, which is the quickest and easiest of the four endings available.
Major spoilers ahead:
*** Spoiler - click to reveal ***After beating Guardian Ape in Sunken Valley and the Corrupted Monk in Ashina Depths, you will need to return to Ashina Castle and talk to Great Shinobi Owl on the rooftop.
After the conversation, you will have to choose if you want to betray Kuro or stay loyal to Kuro. For the Shura ending, you will need to choose the first option to "Obey the Iron Code, forsake Kuro”.
This will trigger two boss fights against Emma and Isshin Ashina. Once you have defeated them both, you will unlock the Isshin Ashina trophy as well as the Shura trophy below.