To find "The End" easily create two worlds with the same seed and different names.
You must be using newer world's that have The End portals already in them for this to work. I believe every update the seeds change so seed 1 may be a bare wasteland and next update seed 1 could be the perfect world with all 23 biomes.
On one world play in creative mode and then take Eye of Enders and throw them and follow the eyes till you find the stronghold where its near. Then mine out the area or use TNT to find the portal.
When you have found it mark its location down so you know where to find it later.
After that start your second world with the same seed on survival. Play the game as you would until you have found some items for a diamond pickaxe, get obsidian, make a Nether portal, find a Blaze spawner, kill Blazes for rods, and then kill Endermen for the Ender pearls.
Also noted if you can, try enchanting an iron or diamond sword with looting. You can save before you enchant it and reload the save as tedious as it can become the enchantment is totally worth it.
You will also want to get a Pumpkin for a helmet as well, Endermen will not teleport away from you if you are wearing a pumpkin on your head which makes them die as simply as anything else.
If you have a few buddies and have a world with no pumpkin seeds or pumpkins what so ever you can make a fishing rod and throw it at an endermen to leash it so he can't teleport to far away and just follow the line till you find him and kill him.
Once you have twelve Ender Pearls make them into Eyes of Ender with the Enderpearls and Blaze powder, place them into the twelve open slots and then you should see the teleport open and be looking into space.
Jump into the portal and the achievement will pop up. If you are prepared then kill the Ender Dragon on like peaceful or whatever you want for the other achievement
"If you are doing it on your own I also recommend saving the game every time you come across an enderman so you will be able to reload that savefile if the enderman didn't drop a pearl and than repeat till he does"
- No Parasite