The Clan Leader and Officers can promote other Clan members.
To promote a member tap their username in Clan view and select "Promote". Regular members will be promoted to Officers. If you are the Clan Leader you can also assign leadership to an Officer by promoting them.
Note that if you as Leader promote an Officer they will immediately take over the position of Leader. Don't do this unless you're absolutely sure you want to give away the control of your Clan.