I think a key thing as spy is to avoid groups completely. If you reduce the chances to bump into someone then it will make life easier.
I think spies work best when they are trying to pick off the people at the back of the lines, whether it's people running from spawn, or engineers or people going back for health. Kill, cloak, hide.
If you're in a situation where, when cloaked, you're trying to get through a group of enemies, either following, or running through the front line to get behind the enemy, then I highly recommend jumping from side to side. Being able to reduce the area of possible contact with an enemy is the key.
When disguised there is a little bit of timing involved, as well as a clear plan. You need to look like you have a definite movement, nothing gives away a spy as someone just running around after another person. If you attack when they are attacking then it's easier to blend in and not look suspicious.
I do my attacks as spy in a forward motion. Cloak, get in behind the lines, disguise, decloak, run forward with the enemy attackers and attack. No matter the result, I push through to my team-mates, heal and resupply. Rinse and repeat. Obviously if your using the cloak and dagger or dead-ringer then it's different.
There is a huge slice of luck involved with tracking and pushing through as a spy as people are a little unpredictable and sometimes you just get unlucky.
As a spy, you're best off spending your time working to assassinate the opposing medic. Since the other team has only one medic, it is a huge advantage to your team if you can kill (or badly hurt) the other medic, even at the expense of your own life.
Since this is a competition-level play, you can expect full communication from the other team. You will very rarely fool them into thinking you're on their team. They all should know where everybody is. (Whether or not they really do is another matter). The simplest way for you to be effective is to find a nice sport to Cloak and Dagger, wait for the enemy medic to run by, and revolver or ambassador them from a distance.
Mix up your spots, be unpredictable, and stay hidden. You may also want to spend some time hounding the enemy sniper, depending on his skill level. I've seen highlander teams completely fall apart from a skilled enemy sniper. On the off chance he's razorbacking, the ambassador will be the weapon you want to go with. Mix your disguises, but definitely use the sniper disguise - that class matches closest with this play-style.
Best Answer
Spawn-camping is one of the riskier strategies a team can employ - you are fighting the enemy team outside a one-way door which only they can open, and behind which is an infinite supply of health and ammo that only they can access. It only works if the team being spawn-camped is so uncoordinated that they would be losing badly anyways even if not being camped. The team inside spawn actually has the advantage of being right next to their resupply cabinets - they can change classes and loadouts and refill ammo and health at will.
Breaking even a serious spawn camp merely requires some coordination. Running out one at a time is not going to work - they will have time to repair their sentries and reset their sticky traps in the time it takes you to respawn. Have a medic (preferably two or three - they probably have at least one medic) build uber (normal uber) on some scouts / soldiers / demomen who can inflict self-damage to speed up the process. Deploy the ubers on some heavies right out of the gate and clear any sticky traps and other immediate threats that may be present. Have the demomen and soldiers follow behind to destroy sentry nests and teleporters. Once you do that there should be no more spawn-camp. Kill as many enemy players as you can and go for the objective to make up for any lost time.
If you are playing on a public server and your team is uncoordinated enough to get spawn-camped, it is probably game over. There is very little you can do alone, unless you are a very good spy (but remember, even if you walk out cloaked, they can see the door open). If you feel up to it, try coordinating / organizing your team a bit via chat (text or voice). Nobody likes being spawn-camped so odds are if someone takes charge people will follow.
One last note - most maps have more than one spawn exit (typically 3). If they aren't camping all exits simultaneously you can just sneak out and let the campers waste their time.