What is the most efficient way to increase my sneaking level early on?
Perhaps not the most efficient method, but this can be done AFK.
In Whiterun, just next to the accessory shop, you can crouch in a corner, behind the Whiterun guard. Hold the movement stick back*, so that you are "sneaking" backwards into the corner of the building. Leave the Console run overnight, and you will have 100 sneak by the next morning.
This is one of many places that this can be done, but basically, you just want to set yourself up in a location where you are in sight of someone, but can't be seen (the "eye" indicator is closed), and walk backwards (this obviously won't work if there isn't someone to sneak past).
Again, while not the most efficient in terms of in-game-time, it is in terms of your time.
User Kallum Tanton points out that the PC version has an "auto run" key (default C). This can be used by facing toward the wall instead.
The best way to level up your sneak skill is to successfully execute a sneak attack - the amount of damage doesn't matter, it's a flat amount of XP per attack. For some easy targets there's a small bandit camp just North of the cave where you exit Helgen Keep - there's 3 bandits there. Also Embershard Mine is good for sneak attacks, it's fairly easy to split up the bandits there individually to prevent them from ganging up on you. Before you leave the cave though, you can execute sneak attacks on whoever your chose to follow (Hadvar or Ralof) - there's no penalty for attacking them and you still get the XP.
Other than that just sneaking around undetected will build up your skills. The more NPC's around, the more XP you earn. Works better with shopkeepers and best with hostile NPC's. If you plan on going to Riften, Goldenglow Estate has up to 9 hostile NPC's roaming the grounds - sneaking around these will massively boost your sneak skill.
If you have some gold you can also use trainers - the most accessible one early in the game would be Khayla - she is a Khajiit with Ri'saad's caravan which travels between Whiterun and Markarth, so keep an eye out for their camp outside Whiterun.
And of course there's skill books, which can be bought or found. The closest one you can find somewhat easily in Whiterun is 2920, Last Seed found in Redoran's Retreat on a crate in the back cavern.
They're the (IMO) best options near the start of the game. For more ways to level up, see this WIKI page.
The best way that got me to 100 really fast was the Greybeards. If you continue through the main story, you come across the Greybeards really fast. Here's how to increase your sneak level: The greybeards are meditating a lot. While they do, you can sneak and backstab them 4 times. don't stab them a fifth time, because they will kill you then. Just wait in-game until they are meditating again or find another meditating greybeard. Backstabbing them will increase your sneak level very fast.