When can I get Covetous Shen's conversations for the Everybody Loves Shen achievement?

Because some of these are only available after the Jeweler has been unlocked before starting A Shattered Crown in act I, you will have to start already completed quests after having unlocked the Jeweler to get his Act I conversations.

The town becomes inaccessible after you start the objective Find Diablo during The Prime Evil in Act IV and so any not heard by that point have been entirely missed.

Act I

• Tell Me About Yourself: Available from the time after you complete the quest The Legacy of Cain and before you accept the quest A Shattered Crown until you are given the objective Find Diablo during the quest The Prime Evil in Act IV.
• The Restless Dead: Available from the time after you complete the quest Legacy of Cain and before you accept the quest A Shattered Crown until you begin Act II.
• Deckard Cain: Available from the time after you complete the quest Legacy of Cain and before you accept the quest A Shattered Crown until you accept the quest The Doom in Wortham.
• Jewels: Available during the quest A Shattered Crown.
• King Leoric: Available from the time after you complete the quest A Shattered Crown and before you accept the quest Reign of the Black King until you complete the quest Reign of the Black King.
• The Fallen Star: Available during the quest Sword of the Stranger.
• The World: Available after accepting the quest The Broken Blade until you accept the quest The Doom in Wortham.
• Dark Visions: Available during the quest The Doom in Wortham.
• Grieving: Available during the quest Trailing the Coven.
• The Stranger: Available during the quest The Imprisoned Angel.
• The Journey Ahead: Available during the quest Return to New Tristram.

Act II

• What services do you offer?: Available from the time after you begin Act II and before you accept the quest Shadows in the Desert until you begin Act III.
• A Mysterious Rumor: Available from the time after you begin Act II and before you accept the quest Shadows in the Desert until you accept the quest Blood and Sand.
• The Desert Heat: Available from the time after you begin Act II and before you accept the quest Shadows in the Desert until you accept the quest A Royal Audience.
• A Special Jewel: Available after accepting the quest Shadows in the Desert until you begin Act IV.
• A Case of Mistaken Identity: Available after accepting the quest Shadows in the Desert until you begin Act IV.
• Caldeum: Available from the time after you complete the quest City of Blood and before you accept the quest A Royal Audience until you complete the quest Unexpected Allies.
• Zoltun Kulle: Available from the time after you complete the quest Unexpected Allies and before you accept the quest Betrayer of the Horadrim until you complete the quest The Black Soulstone.
• Legends and Myth: Available after accepting the quest Blood and Sand until you begin Act IV.
• The Jewel's Price: Available after accepting the quest The Black Soulstone until you begin Act IV.
• The Black Soulstone: Available from the time after you complete the quest The Black Soulstone and before you accept the quest The Scouring of Caldeum until you begin Act III.

Act III

• The Frozen North: Available from the time after you begin Act III and before you accept the quest The Siege of Bastion's Keep until you accept the quest Machines of War.
• Memories: Available after accepting the quest The Siege of Bastion's Keep until you are given the objective Find Diablo during the quest The Prime Evil in Act IV.
• More Memories: Available after accepting the quest Turning the Tide until you are given the objective Find Diablo during the quest The Prime Evil in Act IV.
• Hard Questions: Available after accepting the quest Tremors in the Stone until you are given the objective Find Diablo during the quest The Prime Evil in Act IV.
• The Battlefields: Available after accepting the quest Machines of War until you accept the quest Heart of Sin.
• The Nature of the Jewel: Available after accepting the quest Siegebreaker until you are given the objective Find Diablo during the quest The Prime Evil in Act IV.
• The End: Available after accepting the quest Heart of Sin until you are given the objective Get the Soul of Azmodan during the quest Heart of Sin.

Act IV

• The Hunt Continues: Available after accepting the quest Fall of the High Heavens until you are given the objective Find Diablo during the quest The Prime Evil.
• Hope: Available after accepting the quest Fall of the High Heavens until you accept the quest Beneath the Spire.
• A Dark Day: Available during the quest Fall of the High Heavens
• Leah's Soul: Available after accepting the quest The Light of Hope until you are given the objective Find Diablo during the quest The Prime Evil.
• God or Man: Available after accepting the quest Beneath the Spire until you are given the objective Find Diablo during the quest The Prime Evil.
• Ruin and Corruption: Available during the quest Beneath the Spire.
• The Lord of Terror: Available after accepting the quest Prime Evil until you are given the objective Find Diablo during the quest The Prime Evil.
• Victory: Available after defeating Diablo but before speaking with Auriel to end the quest and game.