How to get better at sneaking

There's no real reason to complete this undetected (except for personal feel-goodsies, perhaps). The mercenaries guarding this place are nameless, "Mercenary" NPCs, so I wouldn't worry too much about killing them.

That said, your best bet is the same way you go about undetected in general - strip naked (armor is noisy!), hit up the dark corners, and make sure to hold shift (the default) to go extra slowly.

Perhaps this is common knowledge, but it certainly tripped me up on my first foray into thieving.

By default, your character is running. All the time. (This is separate from sprinting, which makes you move even faster and affects your FOV.) Unless you prevent it, you even run while sneaking, and this understandably hinders the effectiveness of your sneak.

On the PC, you can stop running by either holding Shift or pressing Caps Lock (by default). Ceasing to run will cause your character to slow to a crawl (or so it will feel) and dramatically improve your stealthiness.