To easily increase your Sneak skill, find someone who is sleeping. Find a wall that they are near and run against the wall while sneaking. As long as no one can see you, your skill will go increase until the person wakes. If this happens, wait until night and let them go back to sleep.
When you have the option of joining the Thieves Guild by meeting Armand Christope at the Imperial Water Front garden around 11 p.m., accept his task. He will offer you the option of purchasing lockpicks. Do not buy any. Go into sneak mode and get behind him. Not only will you get the lockpicks from him, you can also pickpocket him repeatedly to build up your Sneak skill. He will just tell you to go ahead, and that he does not need it anyway.
Whenever you get in jail, go into sneak mode and move around for awhile until your Sneak level rises.
After the assassination of Emperor Uriel Septim (during the opening mission), leave the Emperor's body with one of his guards, Baurus. After walking down the newly revealed "secret" passage, turn around. You can still see Emperor Septim's body and part of Baurus. Walk to the nearest corner (to your left), put yourself in sneak mode, and walk forward. Your Sneak level will increase quickly from here. If you have a wired controller, keep the Analog-stick held Forward and let the game idle. Check back every fifteen minutes, as the game will pause as you reach achieve Apprentice, Journeyman, Expert or Master classifications. If done correctly, you should be at 100 Sneak within ninety minutes.
Join the Dark Brotherhood. When you receive your first quest to kill Rufio, enter his room and go into sneak mode. Since Rufio sleeps for 20 hours a day, walk around in his room to gain very easy Sneak levels.
Complete the Fighting Arena in the Imperial City and become Grand Champion. Once you collect all your money from the Bloodworks person, go outside to find a kid that runs up to you and say that he is your biggest fan. He will ask if he can follow you around. Choose "Yes you may follow me around" and he will go wherever you go. Go to a place away from any guards or people. Crouch into sneak mode and choose "Pickpocket" on the kid. You will be brought into his inventory.
Press CIRCLE to exit out of his inventory. Do not steal anything. Repeat this by pressing X to pick pocket and CIRCLE to exit. You will quickly gain Sneak experience. He does not care if you pickpocket him because he is your "biggest fan".
Go to the Dark Brothers base and find the Rat Schemer, then sneak behind him. You should get easy Sneak experience if no one is in the room since Schemer will not notice you. Note: You must be in the Dark Brotherhood to do this.
Go to the Arena, then bet on a match and go in the door. When you are in the door, go to the right and look for a man on a chair. If he is not there, wait until match is over. Then, go outside and wait for a day. Then, go back and bet on another match and look for the man again. If he is there, walk into the corner behind him and sneak. Keep your character running into the wall to gain experience very quickly and be able to be a master in about an hour.
Go to the Imperial City and get sent to jail. Once in jail, get out by either pickpocketing a guard or by using the Open spell or a lock pick. Once outside the cell, kill the guard and go to the cell next to yours. There should be a man inside. Crouch and pickpocket him repeatedly. Keep doing this until you reach the desired level of Sneak.
At the Arena, go behind and in the corner of Yisabel. Go into sneak mode then walk backwards.
An easy way to increase your Sneak level is to complete the Arena and get the adoring fan. Take him with you next to a bed and pickpocket him. Exit the item screen once he has been pickpocketed, then repeat the process until a message in the top left corner appears, stating that you have to rest. Rest, get your level up as an added bonus, and continue.
An easy way to increase your skill in Sneak is during one of the Dark Brotherhood contracts where you must assassinate a pirate captain. You can get the option of hiding yourself in a crate that is just outside the ship. Hide in a crate and you should soon be inside the ship. Climb up the ladder to the upper level to reach a small room. You may now sneak into a wall and gain Sneak experience. Make sure to check occasionally for the message of an increased mastery level in that skill.