Here are the chapter-driven achievements, just progress through the story to acquire them. They are arranged chronologically.
Albus Percival Wulfric Brian
Off the Beaten Track
Attempt to Resist
Did Santa Eat That Cake?
Accordion to Grawp
Phoenix Rising (Finish Year 5)
The Slug Club
Weasley's Wizard Woes
A Riddle Revealed
I am the Half-Blood Prince (Finish Year 6)
Year 7 - Part 1
Cake or Death Eater?
A Wise Disguise
The Tale of the Three Brothers
Here Lies a Free Elf
To Be Continued (Finish Year 7 - Part 1)
Year 7 - Part 2
Undesireable No. 1
You and Whose Army?
Wit Beyond Measure...
Kick the Bucket
All Was Well (Finish Year 7 - Part 2)
Complete the Bonus Level: LEGO CITY
To unlock the LEGO CITY, complete story mode once and build the door at the upper level of Borgin and Burke's.
But... I Am The Chosen One
Complete the game 100%
Hogwarts has Changed
Visit the Hogwarts Foyer in Year 7
Whenever you can visit the tent in the woods during Year 7, go inside and use the Marauder's map at the left table. This teleports you to Hogwarts using Neville, Ginny and Seamus where you can collect various stuff. This disappears after the endgame, so make sure you get it immediately.
Complete the game 50%
Use a Quibbler dispenser 25 times.
Quibbler dispensers are the boxes that give out SPECS.
Stand still with no controller input for 5 minutes.
Complete all lessons.
We are the D.A.
Unlock all members of Dumbledore's Army
Weasley Does It
Use a Weasley Box with every Weasley
That would be Ron, Ginny, Arthur, Molly, Bill, Charlie, Percy, Fred, and George. Once you've bought all of them, the surefire way to get this is to head to the Great Hall Foyer and use the cauldron and Weasley boxes.
Complete the scene where Harry and Hermione dances in the tent.
Just shoot the radio inside the tent.
Knuts and Vaults
Collect 1 billion studs.
Defeat every Harry freeplay variant as Voldemort.
First up, unlock all Harry and Voldemort characters. Then stand in front of a cauldron and change between the two of them, while casting the Dark Magic spell Avada Kedavra using Voldemort all the time.
Unlock all of the Order of the Phoenix variants.
A Minifig's Best Friend
Unlock every character with a pet.
Unlock all witch characters.
Unlock every pyjama character variant.
A Dish Best Served Cold
Defeat Bellatrix with Neville (Waiter) in a duel.
Unlock Neville (Waiter), then simply use her when you replay any story chapter where you fight Bellatrix as a boss.
Not "Fun Guys"
Defeat 30 Red Caps.
The mushroom critters. They are mostly found in the road path at Hogsmead Train Station and in the Woods.
Dark Times Ahead
Unlock every bad wizard.
Use the Deluminator.
A Sirius Family Issue
Defeat Bellatrix with any Sirius variant in a duel.
Unlock any Sirius character, then simply use him when you replay any story chapter where you fight Bellatrix as a boss.