Take it from a guy who plays Spy very frequently in PUGs, you have to approach organized play as Spy with a completely different mindset, unlike other classes. Your normal way of getting kills won't work very often, so you have to change how you play.
1. Do not go for a backstab on an enemy because he is unaware, go for a backstab on an enemy because he is distracted.
Here's why: An unaware player may suddenly realize that he hasn't checked his back in a while, and many players appear to be unaware, but will randomly turn around and see you. And even if he doesn't, if any one of his teammates see you and call you out, he will instantly become aware, and you will often die as a result. But when a player is distracted, even if several people call you out, he is still distracted and thus will not turn around. Often times players will be too busy trying to dodge incoming fire/hitting the enemy in front of them that they won't even turn around when a Spy is called out. And when the enemy is distracted, there is a far smaller likelihood that anyone will spot you walking towards them, and even if they do, their suspicion might be outweighed by the fact that they are hoping for help from any teammate in the area, whereas normally good players are extremely suspicious and consider Spies one of the only threats able to attack them when in a "safe" area. It is also possible to chain backstabs together while their team is distracted because people are already dying and screaming, and amidst the chaos of communication a Spy warning will probably fall on deaf ears.
2. Do NOT rely on good spots to get kills.
This is a big one. Many Spies will show you how good some spots are to get backstab after backstab, but they don't work in organized play. Sun (aka SubtleArt) and OMFGNinja will emphasize these spots as very useful, and while they are for scouting, vantage points, and the occasional stab, you should NEVER expect to get more than a single kill out of it, if that. Good, organized teams adapt to that sort of thing very quickly. And even if it does turn out to net you a few kills, know that you could be much more effective by attacking distracted enemies near the front. Because in reality when a Heavy that the attacking team was relying on suddenly dies, they are thrown into a panic, both in dealing with the Spy and with the sudden change in the battle that they can't react to. Picking off people coming from Spawn can be helpful, but at the same time the enemies at the front lines are prepared and know what people they have or not, meaning that no chaos is actually insued. That's right, CHAOS. That is what Spies can create more effectively than most other classes, because they really are an X factor when done right. When somebody important (which is almost everybody in 6's and 7's) gets picked off suddenly and out of nowhere, from behind, in the middle of a battle, their team can get thrown into chaos because they suddenly have to adapt to losing someone out of nowhere and at the same time deal with the threat of a Spy from behind as well as the enemy in front.
3. Be unpredictable. No, REALLY. Good players know where you're going to go, they know what Spies usually do, and that includes going in the "unexpected" direction. So instead, go in the TRULY unexpected direction.
There are three options when cloaked and you have an enemy in hot pursuit: Run away as fast as possible, run in the "unexpected direction" of straight towards your enemy, and then there's the truly unexpected direction, which is of course undefinable. Sometimes it's into a random corner of the room for several feet, and then out the door farthest from your enemy. Sometimes it's going deeper into enemy territory, or in this case, "danger." But keep in mind that you need to always be unpredictable, because if you can predict yourself, so can other players.
I hope I did a decent jobs laying out these tips, and keep in mind that these are tried and true by yours truly, but are not laws of science, they are merely tips.