You can get an Arch Duke to spawn in 2 ways.
1. Cause some general mayhem and kill lots of enemies so that your notoriety increases to max level. When the Ark Angel spawns there will still be other enemies around which will make this harder.
2. The easiest way is to play one of the Marshalling activities. Once yo have interacted with the required number of control stones (the number varies between 2 to 4), an Ark Angel will spawn. Once he arrives there will be no other enemies to worry about, so simply use your heavy weapons to lower his health and the switch to melee by holding
and then use
to finish him off.
added this in the comments:
I found it really easy to keep Stone Blasting him when he was low on health and punching him, dunno if you want to add that Blast makes it much easier...