Keep Moving.
Contrary to what you may have thought, it's NOT a god idea to stay in one place for long as Sniper. This is because if you remain in a spot after you get a kill, you'll show up on your victims kill feed, letting him know where you are so he can either snipe you himself or simply head over, knock on your front door, and ask if you could crit his next rocket.
After you get a kill (or if you're spotted and are getting shot at), book it to another ideal vantage point for sniping. This will keep enemies guessing where you are and giving you a developing memory of the maps you play on. Given this does take a bit of map exploration to get down entirely, it's good to keep this in mind when playing offensively or defensively.

Quickscope is Key.
Tell me if this sounds familiar: Your Sniper Rifle has 100% charge and you're about to line up your scope for a perfect headshot when, suddenly, a certain snorting frenchman gives you a firsthand(or should I say "back") experience with his butterfly knife. Most of these deaths are due to unaware Snipers hoping that players simply run into their line of sight before Spies find him. I'm here to tell you that in that battle, Spy tends to win.
To avoid this, keep your scope down until you know it's safe to pull it out for a quick headshot. Don't worry if your scope isn't 100% charged. The only class you can't headshot without the charge is Heavy, who moves fairly slow and a second headshot or team assistance after the first can usually kill them quickly.

Keep Calm and Steady.
It's pretty easy to get carried away in a game like TF2 when mowing down with the Minigun, blasting enemies to bits, or going full W+M1 Pyro on whole teams. For Sniper, however, getting carried away like this throws off your aim, making headshots near to impossible (I can't begin to count how many kills I missed with a quick jerk of my hand). This tends to be what makes Sniper difficult for most people: can't keep their hands steady. ~(o-o)~
When sniping, keep your focus on the other team rather than your kill count (you'll know you're doing it right when your kills stack up without you noticing).

DISCLAIMER: I'm not saying this is the best loadout, but it's worked out for me.
The Hitman's Heatmaker
The Heatmaker gives you a faster charge rate when you get 100% focus, but at the cost of 20% bodyshot damage. I personally found it more forgiving than the Bazaar Bargain (also a valid choice if you're confident in guaranteeing consecutive headshots)

Jarate/Self-Aware Beauty Mark
This is perhaps one of the biggest mistakes a Sniper player may make: thinking a Razorback is a better idea than Jarate or a sidearm.
It's not.
All it does is block one backstab and take up space in your loadout that could have been used to counter the other 8 classes you'll face.
I use Jarate not only to make jokes about my enemy's golden complexion, but also for some coverage as well as assists from my team (I'll tell you, mini-crits are tempting).

The Bushwacka
I carry this weapon to complement the Jarate, which will guarantee a critical hit when paired together. It's a great combo for when you're alone and fighting an opposing player. Only class you can't best with this is the Pyro, who gets bonus fire damage on you when you have the Bushwacka out. Best thing to do is follow Sniping Rule 1: PROCEED TO ABSCOND THE MUK OUT OF THERE!

Basically, just keep moving, don't scope too long, keep your eyes on the prize, forget the Razorback's existance and you'll be golden (at least, your enemies will...)