When can I get Lyndon's conversations for 'To Catch A Thief' achievement?

Solution 1:

Lyndon's Origins (Act I-IV) - Available after completing The Broken Blade (Part 6).
The Unrepentant Rogue (Act I-IV) - Available after completing The Broken Blade (Part 18).
Kingsport Fighting (Act I-IV) - Available after completing The Doom in Wortham (Part 9).
Sticky Fingers (Act I-IV) - Available after completing Trailing the Coven (Part 15).
Difficult Decisions (Act I-IV) - Available after accepting Return to New Tristram (Part 1).
Blood Ties (Act II-IV) - Available after completing Shadows in the Desert (Part 3).
Brothers Divided (Act II-IV) - Available after accepting City of Blood (Part 1).
The Guard's Fate (Act II-IV) - Available after accepting Unexpected Allies (Part 3).
First Love (Act II-IV) - Available after completing Betrayer of the Horadrim (Part 7).
The Thief's Secret (Act III-IV) - Available after accepting The Siege of Bastion's Keep (Part 1).
The Scoundrel's Wealth (Act III-IV) - Available after completing The Breached Keep (Part 5).
Brotherly Love (Act III-IV) - Available after accepting Heart of Sin (Part 1).
A Dark Outlook (Act IV) - Available after accepting Fall of the High Heavens (Part 1).
A Lost Family (Act IV) - Available after accepting Fall of the High Heavens (Part 1).
A Final Gift (Act IV) - Available after accepting Prime Evil (Part 1).

Solution 2:

As all the followers appear before you after you defeat the final boss, almost all of their conversations are available until the end of the game, but there is one that can be missed so I'll mark it with a +. Many of the conversations will unlock only after having heard other previous conversations, but will not become available until the point specified. Also, some of the follower conversations may be hidden if they are not following you, but by having them follow you, you can get a complete list (without the extras).

Act I

Lyndon's Origins: Available after accepting the quest The Fallen Star until the end of the game.
The Unrepentant Rogue: Available from the time after you complete the quest The Broken Blade and before you accept the quest The Doom in Wortham until the end of the game.
Kingsport Fighting: Available from the time after you complete the quest The Doom in Wortham and before you accept the quest Trailing the Coven until the end of the game.
Sticky Fingers: Available from the time after you complete the quest Trailing the Coven and before you accept the quest The Imprisoned Angel until the end of the game.
Difficult Decisions: Available after accepting the quest Return to New Tristram until the end of the game.

Act II

Blood Ties: Available from when you are given the objective Go through the Sundered Canyon during the quest Shadows in the Desert until the end of the game.
Brothers Divided: Available after accepting the quest City of Blood until the end of the game.
The Guard's Fate: Available after accepting the quest Unexpected Allies until the end of the game.
First Love: Available from the time after you complete the quest Betrayer of the Horadrim and before you accept the quest Blood and Sand until the end of the game.

Act III

The Thief's Secret: Available from the time after you begin Act III and before you accept the quest The Siege of Bastion's Keep until the end of the game.
The Scoundrel's Wealth: Available from the time after you complete the quest The Breached Keep and before you accept the quest Tremors in the Stone until the end of the game.
Brotherly Love: Available after accepting the quest Heart of Sin until the end of the game.

Act IV

+ A Dark Outlook: Available during the quest Fall of the High Heavens
A Lost Family: Available after accepting the quest Fall of the High Heavens until the end of the game.
A Final Gift: Available after accepting the quest Prime Evil until the end of the game.

Extras

These don't count toward the achievement

A Worthwhile Quest: Available from when you are given the objective Talk to Auriel and bring your quest to an end during the quest Prime Evil until the end of the game.
A Hero for the Ages: Available from when you are given the objective Talk to Auriel and bring your quest to an end during the quest Prime Evil until the end of the game.
Deeply Moved: Available from when you are given the objective Talk to Auriel and bring your quest to an end during the quest Prime Evil until the end of the game.