At the Varrock Museum, you can talk to him about dragonkin and tell him information about things you've done around Gielinor that involve dragonkin. This includes: completing the quests A Tail of Two Cats and Ritual of the Mahjarrat and killing both the King Black Dragon and Queen Black Dragon (and receiving the First dragonkin journal). Telling him about these things will earn you kudos with the museum.
At the Anachronia Base Camp, you can talk to him about being in the field again, what his plans are, and about the creatures found around the island.
Mr. Mordaut plays the following roles in these quests: