A very easy way to increase your Sneak skill is to find someone who is sleeping. Find a wall that they are near, then 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 allow them go back to sleep.
When you have the option to join 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 to buy 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 repeatedly pickpocket him 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. Move around for awhile and your Sneak level will rise.
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 should still be able to see Emperor Septim's body and part of Baurus. Walk to the nearest corner (to your left), put yourself in sneak mode, then walk forward. Your Sneak level will increase quickly from here. If you have a wired controller, you can keep the Analog-stick held Forward and allow the game to idle. Check back every fifteen minutes, as the game will pause as you reach achieve Apprentice, Journeyman, Expert or Master classifications. If you done correctly, you should be at 100 Sneak within ninety minutes.
Join the Dark Brotherhood. When you recieve your first quest to kill Rufio, enter his room and go into sneak mode. 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. There will be a kid that runs up to you and say that he is your biggest fan. He will ask if he can follow you around. Select "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. Select "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 gain Sneak experience very quickly. 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. Sneak behind him. You should get easy Sneak experience if no one is in the room because Schemer will not notice you. Note: You must be in the Dark Brotherhood to do this.
Go to the Arena. When there, 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 you will 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 you are outside the cell, kill the guard then go to the cell next to yours. There should be a man inside. Crouch and pickpocket him continuously. Keep doing this until you reach the desired level of Sneak.
Go to 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, then 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, then continue.
An easy way to increase your skill in Sneak is during one of the Dark Brotherhood contracts where you are to 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. You should now be in 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.