This will be a journey, so brace yourself.
First of all, I would like to point out that people have made it from the spawn to killing the dragon in just 15 minutes. Look it up. Don't expect it to be this quick yourself. The fastest I've done it is about 6 hours in-game time, but I wasn't really keeping track.
This guide will be for those who aren't worried about rushing to the end. It would be good to do this in a world that you plan on spending time in and getting other achievements in.
Important Note- While working toward the end, always try to kill enderman at night time and get their pearls if you can. This will come in handy later.
Step 1: Play the game normally for a little
Gather food, build a house, go mining; do anything to prepare yourself for the journey ahead. This is a great opportunity to get some of the easier achievements as well (such as opening the inventory).
Step 2: Go to the nether
You need blaze powder in order to craft eyes of ender. Blaze are found primarily in nether fortresses, which are made of maroon colored bricks, but you can find blaze anywhere in the nether. They drop blaze rods, which each make 2 blaze powder. You will need about 15 blaze powder in order to get to the end, so try to get at least 8 rods.
Step 3: Enchant a looting sword *optional*
The other ingredient for eyes of ender are ender pearls. Endermen alone are somewhat uncommon, and some of them don't even drop any pearls. To turn the odds in your favor, you will need to enchant a sword with any looting enchantment. The higher the looting level, the better. You will need both experience and lapis lazuli to enchant an item. The way enchanting works is that the given enchantments are set to match the item you put in. If you enchant something, the given enchantments reset and you have different options. You will be able to see at least one enchantment for each of the options, but they can have more. Getting the looting enchantment is entirely luck based.
There are two ways to do this: enchant the sword directly, or enchant a book and then apply it to the sword. This can be useful because if the sword does not have the option for looting, the book still might. Therefore, it would be best to check the options of the sword first, then a book. If neither have looting, the best option would be to enchant a different item with a level one enchantment so that it resets, then check again. You may need to do this repeatedly in order to get looting.
If you are unlucky and are running out of experience, there are several ways to get more fast. Mining quartz in the nether on peaceful difficulty can yield a good amount of experience as there is a lot of quartz. Killing mobs at night time and mining in the overworld will help too. if you find a spawner, you can also set up an xp farm. Look on youtube on how to make an efficient one.
Step 4: Kill the endermen
Now that you have a looting sword (if you decided to make one) it's time to kill some endermen. The best way to do so is to aim at their legs and attack them with a sword. If you do this, they won't teleport away. Arrows will not hurt endermen and they will automatically teleport. With a strong looting enchantment, a single enderman can drop up to 4 or 5 pearls. You need to get about 15 pearls just to be safe.
Endermen hate water, so avoid lakes, rivers, and the ocean. They will also be impossible to find and kill while it is raining outside. They do not die from sunlight, but they will despawn if they teleport while it is daytime.
Step 5: Find the stronghold and portal
Now you have to use the blaze powder and the ender pearls to make eyes of ender, which are critical to your mission. You need 12 to activate the portal. However, you should always make extra because if you throw one in the air, there is a chance it will disappear. Generally, it will fall back to the ground though. Throw an eye into the air and it will fly in the direction of the stronghold. Pick up the eye if it fell and head in that direction for awhile, occasionally throwing eyes to find your way. When you are there, the eye will not fly as far as normal, or just fly straight up, and fall back down. If you've gone too far, the eye will fly the other way.
Once you are there, you have to dig a hole (preferably with stairs so you can return to the surface) down into the stronghold. You will know if you have reached it when you hit a wall of stone bricks. Strongholds can be very large and will likely have a ton of mobs in them. A unique mob is the silver fish, which spawn out of random stone brick blocks when broken. You will need to search through the stronghold to find the portal room.
The portal room will have a silverfish spawner in front and a staircase leading up to a tan/navy blue portal. Destroy the spawner with a pickaxe and cover the lava with some blocks. I usually put a bed in this room and sleep so if I die in the end, I don't have to run all the way back. To activate the portal, place your remaining eyes of ender in the open slots. If you are lucky, there will already be some slots filled (this is completely random). Now you just have to jump in and you will enter the end.
Note: If you want to collect dragon's breath, be sure to bring a glass bottle!
I will post a guide on killing the dragon under the related achievement.