Thief

There are 53 objectives in the game, plus another 4 in the DLC. Without replay you can only get 36 of them (40 with DLC) in a single game. The objectives, confusingly referred to as "Challenges" occasionally in-game, are shown in your menu under "Objectives" during a mission. You can review the ones that have been awarded by looking in your menu under "Player Progression -> Statistics".

The list of objectives by mission is at the bottom of this solution. Here's some detail on how they work:

Chapters
In main story missions - "chapters" - there are four shown, but you can only be awarded two of them. There is always a "Collect All Loot" objective, and it is always obtainable. There are also three others, and you will be eligible for one of them at the end of your mission. Which one you are eligible for depends on your playstyle.

As you play through each level, the game decides what "style" your play is, based on how stealthy you are ("Ghost"), how many objects you interact with ("Opportunist"), and how many people you kill or knock out ("Predator"). At the end of the level, you will be awarded the Thieving Objective for your playstyle if you satisfied its requirements. It doesn't matter if you filled the requirements for the other playstyles, they don't count. Also, if you manage to perfectly balance two separate playstyles, you can be awarded the "Hybrid" playstyle, and miss out on all of them. The only way to avoid this is to heavily favor one play style, or to replay the mission. Alternatively, you can forgo that challenge and use the slack available to still get to 25.

Client Jobs
In addition to the chapter challenges, there are also 20 more available in client jobs from Ector and Vittori. These challenges do not depend on playstyle.

For most of the six client jobs, there are four objectives available. The final mission from Ector only has one, and Vittori's second job has two that are mutually exclusive. Details on how to start these missions are below with the objective list.

Strategy
If you aren't going for every piece of loot in every chapter for your own reasons, there's no reason to go through the pain of trying for that objective more than once. You'll need to do it once for another achievement:

After that, you can safely omit the "Collect all loot" objective from each chapter mission, greatly speeding things up. You'll have 9 objectives completed from the chapter missions, and you can easily get the other 16 from client jobs. Each client job is short, and they have no separate areas - you can always backtrack throughout the job to find any missing loot, so "Collect All Loot" is much simpler and far less tedious than it is in the chapter missions.

It's not a bad idea to refrain from upgrading your bow damage until you've finished the first five of the client missions. There are two client missions that have both combat takedowns and remaining undetected as objectives. It's possible to get both in one run; pull out a broadhead arrow and fire it into the knee of an unsuspecting guard. While they contemplate transferring to Skyrim, walk up behind them and perform a takedown. If you aren't getting the takedown prompt, try swinging your blackjack at thin air once - away from the guard - that always worked for me. The problem with the increased bow damage is that it can kill the guards outright, making this problematic. Trust me, it's no fun to try to down guards with repeated blunt arrows while trying to stay hidden.

If you are paying attention to this achievement from the beginning, it should be no issue to get 25 objectives as you play. If you miss one while playing through, I'd recommend pushing forward rather than immediately replaying the chapter.

If you are forced to replay parts of the game, I recommend replaying the client jobs rather than the chapter missions - they're far shorter, easier, and have four objectives available rather than two.

Replaying
As mentioned before, you can only get 36 of the challenges completed without replaying missions. You can, if you want, replay missions to boost your objective count. You cannot get the same objective over and over again. You can, however, replay to get an objective you missed, or to get the objective tied to a different playstyle than you got the first time through.

To do this, you must return to the mission's beginning point in the city map and start your replay there. Do NOT use the "Replay" option at the end of the mission. Also, be aware that the game's counter will no longer be accurate. Here's the details on how that all works, hidden behind spoiler tags to save room:


CREDITS
Thanks are due to several TA users whose comments on this solution dramatically improved it. lTN95l set me straight on the client objectives in the first place. It was IL1971 that pointed out that two of the challenges in Vittori's second job are mutually exclusive. RoiDesFeuilles and xxDrunkenJedixx both discovered the "Hybrid" playstyle - much to their dismay, I imagine. BrokenKnight was the source of the DLC information. Finally, iAMxYUNGxGxBABY helped iron out the intricacies of the replay system.

Chapter Challenges (main story)

Predator Challenges are marked with (P), Opportunist with (O), and Ghost with (G).

Chapter 1:

Chapter 2:
Chapter 3:
Chapter 4:
Chapter 5:
Chapter 6:
Chapter 7:
Chapter 8:

Ector Challenges

Note: If you haven't met Ector yet, here's how:


Ector Job 1 (must be past Chapter 1):

Ector Job 2 (must be past Chapter 3):

Ector Job 3 (must be past Chapter 6):

Vittori Challenges

Note: If you haven't met Vittori yet, here's how:


Vittori Job 1 (must be past Chapter 2):

Vittori Job 2 (must be past Chapter 4):

Vittori Job 3 (must be past Chapter 5):