The first three pieces of the armor set are unlocked by increasing your Dublin Renown to max level. The final two pieces are unlocked by defeating the two Lost Drengrs in Ireland.
Timestamps:
0:00 - Dublin Champion Bracers (Dublin Renown Level 2)
0:00 - Dublin champion Cloak (Dublin Renown Level 4)
0:00 - Dublin Champion Helmet (Dublin Renown Level 5 / Max Level)
1:31 - Dublin Champion Breeches (Lost Drengr in Connacht)
2:10 - Dublin Champion Armor (Lost Drengr in Ulster)