There are numerous characters throughout the game that can become agents, however I believe only those agents that qualify as perks count to your total; still you will likely need more than 10. It unlocked on number 11 for me. Many involve quests, which will be enclosed with quotation marks below.
Witty Ritts: Requires "Strange Bedfellows" to encounter her. Fight off the Templars engaging her, after 2 examinations you can question her. To recruit her either play as a dwarf or defer to Varric.
Speaker Anais: Complete "Praise the Herald of Andraste." She can be recruited for Cullen or Josephine
Enchanter Ellendra: Complete "My Lover's Phylactery." Either Vivienne must be in party or have the Arcane Knowledge perk.
Lord Berand: Complete "Love Waits." He can be recruited for Cullen or Josephine.
Corporal Vale: Complete all of the following: "Apostates in Witchwood," "A Healing Hand," "East Road Bandits," "Hunger Pangs," "In the Elements," and "Templars to the West." Speak to him after all these then say Recruit the skilled.
Smuggler Tanner: Complete "Blood Brothers" to acquire "Business Arrangements." Defer to Cassandra when speaking to her.
Clemence the Tranquil: In the Redcliffe Tavern. Just speak with him and he will eventually offer his services.
Horsemaster Dennet: Complete "Farmland Security," and "Trouble with Wolves." Then defer to either Cassandra, Dorian or Vivienne or use History Knowledge or Underworld Knowledge.
Barter by Belle: After speaking with the Chantry head find Belle the merchant. She will offer her services.
The Storm Coast
Blades of Hessarian: Start "Cleaning House" to find their base, but first craft Mercy's Crest to challenge the leader; defeat him. The rest of the group will work for you.
Sky Watcher: Talk to him after closing a fade rift he's monitoring. Complete "Lost Scouts" and he'll be waiting outside; you can ask him to join.
Therinfall Redoubt Only available if you side with the Templars
Ser Barris: During "Holding the Great Hall" never let the meter dip below 50%.
Emprise du Lion
Michel de Chevin: Talk with him outside the village first and he will assist in claiming the Keep; it's boss must be defeated to recruit him.
Jana the Recruit You will encounter her on the way to the village. Talk her out of enlisting in the Grey Wardens and serving the Inquisition instead.
Loranil: Complete "By the Grace of the Dalish."
Fairbanks: Complete "Victims of War."
Frederic of Serault: Defeat the Abyssal High Dragon.