You can begin unlocking alternate endings after you went through the storyline normally and unlocked the natural ending for the game. Once you do this, you will get the Fragment Skill "Paradox Scope." The Paradox Scope allows you to be able to live through realities which should have never come into play. You can learn some interesting facts from these endings or just see some fun stuff!
To start with a Paradox Ending, you must find a Gate Seal so you can close a gate. After closing the gate, turn on the Paradox Scope skill to change the future then proceed through the world.
Note: You can get the Fate and Freedom and Heir to Chaos endings in the same go after unlocking the 4xx AF gate. Just get the Fate and Freedom ending then select the New Bodhum area in the Void Beyond. Say no to Lightning and proceed until you can get the Heir to Chaos.
A Giant Mistake
This is an ending you can get in Bresha Ruins 005 AF. In the main storyline, you first weaken Atlas. In this ending, you must kill Atlas without weakening him. It is recommended you have a good medic and sentinel, or are just really high level. Be sure Noel and Serah both have around 3500 HP or more. Once you beat him you'll unlock the Paradox Ending and snag the "Transcript: A Giant Mistake" fragment as well as 11,000 CP.
This ending comes from Oerba 200 AF. For this, you must beat Caius with the Paradox Scope on. Thankfully, you do not need to go through any Temperal Rifts this time; just head staight for the Deserted Schoolhouse and fight Caius. Your reward will be the "Transcript: Vanille's Truth" fragment and 10,000 CP.
Mischievous Mog's Marvelous Plan with Flan
Visit the Sunleth Waterscape 300 AF. For this ending, you simply need to defeat his Royal Ripeness. Once you beat him, you'll unlock the ending and snag the "Transcript: Mog's Marveous Flan Plan" fragment and 10,000 CP.
Visit Augusta Tower 200 AF and proceed to the very top floor and fight against Proto fal'Cie Adam. He is a much stronger version of the Proto fal'Cie Adam you faced before. After defeating him, you get the "Transcript: Test Suvjects" fragment and 10,000 CP.
Beneath a Timeless Sky
You must visit the Void Beyond. Problem is, you cannot just go there by selecting it on a map. You must unlock the gate at Academia 4xx AF. Go through until you give the cores to Alyssa. Hand all five over and accept the Gate Seal from her. Activate the Paradox Scope and continue through the Void until you fight Caius with only Serah. This is probably the most difficult of Paradox Endings to get as you only have Serah and a monster to fight with and Caius is much, much stronger than he was before.
Caius tends to focus ALL his attacks on Serah, unless you have a Sentinel. So, my suggestion: Make Serah a Sentinel and have her provoke Caius. Get a very powerful Commando type monster to do the damage. If needed, switch your monster to a medic and heal, or use potions on Serah. You can also do the reverse, but if you have a good Commando monster, it will kill much faster than a Commando Serah will.
Once you defeat him, you'll get the "Transcript: Beneath a Timeless Sky" fragment and 10,000 CP.
Fate and Freedom
Same as the previous ending: You must unlock the gate at Academia 4xx AF. Go through until you give the cores to Alyssa. Hand all five over and accept the Gate Seal from her. Keep the Paradox Scope off until you continue through the Void until you make it to New Bodhum as Serah. (Turning it on can unlock the Beneath a Timeless Sky ending.) You get the option of saying yes or no to Lightning. Agree to stay with her to unlock this ending. You'll get the "Transcript: Fate and Freedom" fragment and 1,500 CP.
Heir to Chaos
Same as the previous ending: You must unlock the gate at Academia 4xx AF. Go through until you give the cores to Alyssa. Hand all five over and accept the Gate Seal from her. Keep the Paradox Scope off until you reach A Dying World where Serah searches for Noel. Turn on the Paradox Scope before Noel's singular battle with Caius. Noel must defeat Caius and Caius is powered up compared to last time. When you beat this battle, you get the "Transcript: Heir to Chaos" fragment and 10,000 CP.
The Future is Hope
Unlock the 4xx AF gate. Proceed until you give Alyssa five Graviton Cores. DO NOT SKIP THE SCENE and reject the Gate Seal Alyssa tries to present you with. Afterwards, you'll unlock this Paradox Ending. You'll also get the "Transcript: The Future is Hope" fragment and 3,000 CP.