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.
Play style
The play style to take advantage of the Ambassador is something like a behind-enemy-lines sniper. The best way to think about is is as a long-range back stab, since it requires precision to use effectively, but when you are successful, you will take down enemies very quickly.
Ambassador = alternatives
Per usual, your goal as a Spy is to sneak behind your enemies. Normally, your next goal, apart from sapping the Engineer's structures, would be to get in position to back stab your enemies, but with the Ambassador, you have the option of trying to get in a few quick head shots instead, which lets you attack your enemies from further behind than normal, allowing you to play more cautiously than normal as well.
Ambassador vs Revolver
Since the Ambassador emphasizes head shots, and since head shots are easier to get on enemies who are unaware of your presence, the Ambassador is more suited for initiating conflict in situations where a back stab attempt is unfeasible or risky, such as when you come across a sniper equipped with the Razorback, or a Pyro. It would be better thought of as a weapon suited for 'back-of-the-head' shots rather than one simply suited for head shots.
The basic Revolver is more suited to ending conflict by finishing off survivors of failed back stab attempts, or holding off enemies who are pursuing you. Also, since it is normally more powerful than the Ambassador, it is also better for finishing off the Engineer's structures, especially after you have begin sapping them.
Summary
If you are successful at sneaking behind enemy lines but have difficulty in getting up close for back stabs, the Ambassador can be an effective alternative.
Best Answer
Having clocked 538.5 hours as Spy, I think I'm well qualified to answer this question.
The Spy has 3 cloaking devices and that all 3 have different uses:
1. Invisibility Watch
The Invisibility Watch (or its cosmetic reskins The Enthusiast's Timepiece / Quäckenbirdt) is the basic, no-frills watch.
Cloak Time
It has the longest cloaking time, but also takes the longest amount of time to recharge if you're not picking up metal boxes.
Knowing the map layout and ammo drops in particular is essential to using this watch.
Decloaking and Detection
This cloaking device uses the standard decloak sound, and can be heard by enemies at close range.
While cloaked, you can be seen as a translucent outline of your current disguise when you are touched by an enemy or take damage.
Cloak restore amount:
2. Cloak and Dagger
The Cloak and Dagger only drains cloak when you're moving.
Cloak Time
It has slightly less cloak time than the Invisibility Watch. The Cloak meter is automatically restored over time while you stand still under cloak (or with cloaking turned off).
As of the July 2nd 2015 patch (Gunmettle), the Cloak meter can now be restored by picking up ammo boxes whilst decloaked. Cloak gain from ammo pickup is 35% less when compare with Invis Watch.
Decloaking and Detection
This cloaking device uses the standard decloak sound, and can be heard by enemies at close range.
While cloaked, you can be seen as a translucent outline of your current disguise when you are touched by an enemy or take damage.
Usage
The C&D is the best watch for reporting enemy activity over voice chat and picking off stragglers. Just don't pick off too many, or they'll catch on and start spy checking.
It also pairs well with the Your Eternal Reward knife, as it makes finding a single target to backstab much easier, especially so when you're on RED on a Payload map. If the other team does not watch their backs, you can pick off everyone pushing a cart.
3. Dead Ringer
The dead ringer drops a fake corpse and activates cloak when you take damage.
Cloak Effects
The DR can only be used at 100% cloak meter, and when you decloak, you will have a maximum 40% charge. When activated, you receive 65% damage resistance as well as a speed boost for 3 seconds.
Depending on your current disguise, the corpse it drops is different:
Note: As of the April 25, 2016 Patch, the fake corpse is no longer missing a Spy head, making it harder to determine whether a Spy was feigning death.
Decloaking and Detection
This cloaking device uses a loud zapping sound as its decloak sound, and can be heard by enemies at medium range.
While cloaked, your cloak is "perfect" for the first 3 seconds. After that, you can be seen as a translucent outline of your current disguise when you are touched by an enemy or take damage.
Usage
The Dead Ringer is particularly handy if you're BLU on a Payload or Attack/Defend Control Point map to make the enemy waste any sticky traps or Charged Sniper shots.
The DR is exceptionally good to use when pushing carts, as the cart refills your cloak meter very quickly.
Knowing which passages are used less on a map is a huge bonus, such as the inner track on koth_nucleus on the lower level.
Cloak restore amount
Note: The 'Saharan Spy' set used to give you silent decloaks with every type of cloaking device, however all 'Item Set bonuses' were removed as of the July 10, 2013 Patch.