I'm playing this game right.

Am I supposed to be killing people left and right? I'm still in the early stages of the game. I haven't even assassinated the first overseer. In fact, I'm at the part where I have to bypass the first two light walls. This is the first assassination mission. I'm not too good at the sneaky.

The last mission I did was for that crazy old lady, where I had to go into the distillery district and poison the still. I didn't do it. When I looked at the list of people who were getting the elixer next I noticed the dude I rescued from the thugs was on it and I didn't want him to die of the plague. BUT, I had to basically kill everyone in the distillery since I couldn't figure out how to sneak through it.

Does anyone have any pointers they can give me? I'm really enjoying the game but I'm just not so sure I'm even playing it right.

Also, I got the Knife of Dunwall DLC, it has to be played seperately? Is the other DLC worth it? Also how do you assign things to the hotkeys?

I'm on PC BTW.

There isn't really a 'right' way to play. I've played the game multiple times, sometimes killing people left and right like you say, sometimes killing just the people who I decide 'deserve' it, sometimes killing no one. Sometimes I'm a ghost, sometimes I'm as loud as possible, sometimes I'm somewhere between that. So play however you enjoy it, and then do it again a different way!

If you're looking for tips to stay hidden...Hm. Well, blink is your friend, and try to stay above the enemies whenever possible.

Stealth in Dishonored isn't so much about hiding as about movement. Blinking rapidly from place to place can be more effective than sneaking - I encountered paths and doors than I couldn't possibly sneak to unseen, but I could blink and quickly move through them. Upgrade Blink as soon as possible, it's the most useful power in the game.
Think vertically: go up or go down, break the line of sight. The outside of the Distillery is a perfect example: rooftops and pipes take you from the entrance gate all the way to the building. Unfortunately, once you're inside it's not so easy anymore, as you already discovered.

It might also be a good idea to get the first tier of Shadow Kill. Sometimes eliminating one or two inconveniently placed guards can make the sneaking path a bit easier without attracting unwanted attention.

It will get easier when you find more runes. I was stumbling and experimenting in that first mission too, but eventually I learned that there's always an alternative path - you just need to find it.

As for hotkeys, open the skill wheel (middle mouse button), highlight the power or item and press the hotkey.