Some good tips above. To round it all out, here are some more:
1) Get each sneak perk as it comes available, till you get sneak 4 at level 23. That point at level 23 is the holy grail to a stealth build. Up till then, you have to 'walk-sneak', which is slow and annoying as hell, and gets you killed easily if you aren't real careful how you do it. Once you get the sneak 4 perk, you can 'run-sneak' without making any noise, so you can move in fast and attack and then hide a lot faster, without being heard and caught.
2) Use military fatigues, or minuteman uniform, as they will give you good stats and let you wear leather arm and leg pieces on top for more stats and whatnot. Get a point or two in the armorer perk, so you can make it shadowed armor- this adds to your stealthiness. Don't wear heavier armor, it makes more noise. Save up your caps and check out all the vendors you meet, some of them have legendary armor pieces that add nice stats in the agility/endurance/perception department. Those pieces can be modded to be shadowed, too.
3) Learn to use the stealth meter at the top of your screen. Keep one eye on it at all times while sneaking around an enemy. If the brackets start moving in towards the word in the center, whether it is 'hidden' or 'caution', then an enemy is starting to become more aware of the danger approaching... stop for a moment, as you are harder to spot when motionless. You'll soon get used to working that meter, sneaking, stopping, moving again, stopping, using cover between you and them, using shadows, etc., in order to get close enough to do what you need to do. And try to only move towards an enemy when they are facing away from you, if possible.
4) Play during game's night. Sneaking in the daylight is a deadman's game, until you have a lot of good gear and perks later on, anyway. I spend 95% of my gametime during the night cycle in the Commonwealth, just for that reason. You are WAY easier for them to spot between 6 am and 6 pm.
5) Adjust your early expectations. Later on, if you build up right, you can sneak up on a legendary deathclaw and steal its lunch money- but before you get to level 23 or so, you'll have to be a lot more patient, careful, and cautious. You won't be able to get up as close before being seen or heard, even if you do everything right. That's just the way it works. But if you do it right, you should hardly be dying at all.
6) Lose the follower! Or make them stay put somewhere safe when you are stealthing up on something. Nothing will give you away quicker than a dang follower.