Update: This achievement is for fully completing the BASE GAME. If you have any of the dlc installed, it will make it a bit harder since some items have harder ways to unlock, like Godhead, and there's more items in the item pool so it'll probably take longer to find just the items from the base game. While I highly recommend getting all the dlc, if you're super focused on getting every achievement, completing the base game first will make it a little easier on you.
100%ing the game consists of 2 parts: seeing every ending and picking up every item. Your ultimate goal is to completely fill out the items, endings, and secrets pages under "Stats" on the main menu.
There's no real going out of your way for the endings, you'll get them as you play through the game, except for one that takes a little more effort.
*** Spoiler - click to reveal ***Epilogue is unlocked by beating Mom for the first time
Endings 1-11 are unlocked one at a time, each time you beat Mom's Heart
12 is for beating Satan in Sheol
13 is for beating Isaac in The Cathedral
14 is for beating the Blue Baby in The Chest
15 is for Beating The Lamb in the Dark Room
16 is the only one that takes special effort to reach, as it's for beating the secret boss Mega Satan. For this, you first need to beat Blue Baby or The Lamb for the first time, and you will get a message saying "the angels are waiting". From there you must enter two Angel Rooms (alternate Devil Rooms) in a single run. In each, you bomb the angel statue and fight an angel miniboss. They will drop Key Pieces 1&2 (when you collect both you unlock Dad's Key). THEN you get all the way to The Dark Room or The Chest, and the giant door will open, leading to Mega Satan. If you die in the fight, you can open his door again either by collecting the two Key Pieces again or by using Dad's Key.
Now for getting all the items in the game. First, you have to unlock them all.
Each of the challenges unlocks an item. For tips on how to unlock/beat them, check here:
http://bindingofisaacrebirth.gamepedia.com/ChallengesAside from that, you will need to beat every boss and clear boss rush with every character. You can easily track your progress on the character select screen. On the top left, you will see this box:
The symbols appear as you beat the corresponding boss with a character:
*** Spoiler - click to reveal ***The heart on top is for beating Mom's Heart/It Lives
The cross on the left is for beating Isaac in The Cathedral
The star in the center is for beating boss rush (which is opened by beating Mom withing 20 minutes)
The upside down cross on the right is for beating Satan in Sheol
The bottom left symbol (The Polaroid) is for beating Blue Baby in The Chest
The bottom right symbol (The Negative) is for beating The Lamb in The Dark Room
and red outlines indicate you beat them on Hard Mode
Only The Lost has to beat all bosses on Hard Mode, everyone else only needs to beat Mom's Heart/It Lives on hard to unlock a co-op baby for each. (do not even attempt to play as The Lost on Normal Mode as it will not unlock anything)
So, now you've manged to unlock all the items in the game. Now what?
Picking them up of course! Head on over to the Items page under Stats on the main menu to see everything you've picked up at some point. Anywhere you see a question mark, you haven't found that item yet. You can use this page to figure out what you are missing:
http://bindingofisaacrebirth.gamepedia.com/Collection_Page (click on an item for more info)
You will also need to become both of the game's transformations at some point.
Guppy:
http://bindingofisaacrebirth.gamepedia.com/Transformations#G...Lord of the Flies:
http://bindingofisaacrebirth.gamepedia.com/Transformations#L...You will ALSO need to create level 4 Meat Boy and Bandage Girl. These are probably the toughest and most annoying things to get. Tips here:
http://bindingofisaacrebirth.gamepedia.com/Cube_Of_Meat (works the same for Ball of Bandages, Famine will randomly drop either Cube of Meat or Ball of Bandages, and the rest of the horsemen will also drop the same for the rest of the run (except Conquest because he is a jerk)) Getting a level 4 Bandage Girl also unlocks The Bandage.