I suggest you go achievement hunting after completing all the main story missions since you'll be able to freely use unlocked characters on your second playthrough. There are no missable nor bugged/glitched achievements in the game so with enough time and patience, you can reach 100% game and achievement completion.
Gorilla Thriller - 20G
Climb to the top of Wayne tower while riding a Gorilla and playing as a female character.
You can do this using any female character. You need to ride a gorilla from the Gotham Zoo and scale the Wayne Tower via a climbable path along the wall. Watch the video below to learn how to do it:
It's gorilla thriller time!
Kal-El Last Son of Krypton - 20 G
Defeat Zod as Superman
Refer to the video below to learn Zod's location. In case you defeated him using a different character, simply play as Superman and switch the second player as Zod. Attack Zod using Superman until he's destroyed to get this achievement.
Zod is your master now!
Green Lantern's Light - 20G
Defeat Sinestro as Green Lantern
Green Lantern is automatically unlocked after completing all story missions. Sinestro can be unlocked by activating the Metro Station Central bat terminal and defeating him. Refer to the video below to learn Sinestro's location. In case you defeated him using a different character, simply play as Green Lantern and switch the second player as Sinestro. Attack him using Green Lantern until he's destroyed to get this achievement.
Green Lantern Light!
It's A Bird... It's A Plane... - 20G
Fly with Superman
When Superman joins your party in Chapter 5: Chemical Crisis, simply fly by pressing the jump button twice.
Inferior Machines - 20
With Brainiac, defeat any LexBot
You can unlock Brainiac by activating the Gotham Park terminal. You need to destroy the golden lock with Superman's heatray vision or Cyborg's laser. Lexbots appear randomly throughout Gotham City but they do appear frequently on top of the Wayne Tower. Alternatively, you can simply use a cheat (W49CSJ) to unlock the Lexbot. Play as Brainiac and switch the other player to Lexbot and destroy him to get this achievement.
To the scrap pile you go
Girl Power - 20
Unlock all female heroes and villains. (Single Player Only)
The last character token is Supergirl, which will appear from the golden gate that requires all 250 gold bricks in your possession.
The list of female characters in the game are the following:
1. Harley Quinn
2. Poison Ivy
4. Wonder Woman
8. Black Canary
The House of Luthor - 20
Obtain more than 10,100,000,000 Studs (Single Player Only)
Getting this insane amount of studs is very easy, as long as you have all the multiplier bonuses enabled. Refer to the Red Bricks guide to learn how to collect the multipliers. After getting them, make sure to enable them all in the Extras Menu to multiply every stud collected by x3840.
Combo Hero - 20G
Do a finishing move
Enter a fight with 5 or more enemies. After defeating four of them, you'll see a "B" appear above their heads. Press it quickly and you should be able to perform a finishing move.
Subway Hero - 20G
Use the Gotham City Metro
There are three metro stations in the city. You'll know this since they have flat round surfaces and the statue holding those plates rise from the ground when you're near. Simply enter any of the stations to unlock this achievement.
Alternative mode of transport for poor superheroes
Super-Villain - 20G
Unlock all the Bosses (Single Player Only)
Bosses are characters that appear on the map after activating bat terminals. After activating a terminal, their exact location will be briefly shown in and their general location will be marked on the map. Take note that some of these terminals can only be activated by destroying the golden locks binding them. The Yacht Club terminal however, specifically needs the Flash to be rebuilt. The bosses are the following:
Ra's Al Ghul
Toy Gotham - 20G
Complete the Bonus level
To complete the mini-game, you must achieve 1,000,000 studs. Your multiplier studs will be disabled by default. You may want to refer to the series of videos below to know how to complete this bonus level with ease.
Justice League - 20
Unlock all Justice League characters (Single Player Only)
The Justice League members are the following:
You'll unlock most of the Justice League characters by simply completing the main story. Out of the list, you'll need to rebuild the golden gates for Martian Manhunter and Aquaman.
Dynamic Duo - 20
Play a level in co-op
(This must be done by Replaying the Story, not Free Play!) For this achievement, I suggest playing chapter 13: Core Instability. Simply turn on the second controller and leave it there until the level is completed.
My Hero- 50
Rescue all Citizens in Peril (Single Player Only)
There are a totla of 50 citizens in peril; 15 of them are found in each of the story levels while the others are scattered throughout the map. Refer to the Civilians in Peril section of this guide.
Super Hero - 50
Get Super Hero in all levels (Single Player Only)
To achieve the "Superhero" status in each story level, you must acquire a required number of studs. (Usually amounting to 100,000 ) This is easily done if you have multipliers active. (up to x3840) To learn how to get these multipliers, refer to this guide's Red Brick section.
Team Building - 50
Unlock all characters (Single Player Only)
There are 50 characters in the game. Some are unlocked automatically through story completion, others like bosses need to be discovered and defeated, while the others must be purchased from golden gates. These golden gates only become available if you reached the required number of gold bricks. If not, the gate pieces will remain transparent and will display its required number of golden bricks.
The last character token, Supergirl requires all 250 golden bricks to be unlocked. Refer to the Golden Brick and Character Token guides for more details.
Minikit Hero - 20
Use all the Minikit vehicles
There are 15 minikit vehicles (5 each for air, land and sea) which can be assembled by collecting all 10 minikits in every story level. For them to count at "used", you need to press Y and hop on them. The achievement will unlock on the last vehicle you'll hop on.
Test Hero - 20
Test a custom character
You can only create a custom character while inside the Batcave. Head to the left portion of the cave and you'll see two Batman symbols on the wall. Alterinately, you can also press and hold Y to open the character selection screen and choose one of the empty custom character silhouttes.
Extra! Extra! - 20
Collect all the red bricks (Single Player Only)
There are 20 red bricks in the game which can only be collected by having the Joker, Lex Luthor and the Riddler. Refer to the this guide's Red Brick section for the location of each red brick in the game. When scanned on the map, they appear as red dots.
The End - 70
Get 100% (Single Player Only)
You need to meet the following requirements to get 100% in the game:
Find and collect all Gold Bricks (refer to the Gold Bricks section of the guide)
Find and buy all red bricks (refer to the Red Bricks section of the guide)
Unlock and purchase all characters in the game (refer to the Team Building achievement and Character token section of this guide. )
Find and buy all vehicles in the game.
Halfway Through - 50
Get 50% (Single Player Only)
(refer to The End achievement)
City Slicker - 35
Collect all the gold bricks (Single Player Only)
Most of the gold bricks you'll find are scattered throughout the map. (refer to the Gold Bricks section of the guide) The breakdown of other sources of gold bricks are the following:
Completing all story mode levels (15)
Getting the "Superhero" status in each level (15)
Collecting all 10 minikits for each level (15)
Rescuing all citizens in peril (50)
Completing the Bonus Level (Lego Gotham City) (1)
The following achievements are story-related and therefore cannot be missed:
Heroes Unite - 25
Tower Defiance - 25
Core Instability - 25
The Next President - 25
Underground Retreat - 25
Down to Earth - 25
Research and Development - 25
Destination Metropolis - 25
Unwelcome Guests - 25
Chemical Signature - 25
Chemical Crisis - 25
Asylum Assignment - 25
Arkham Asylum Antics - 25
Harboring a Criminal - 25
Theatrical Pursuits - 25