When it comes to the cloak, a good offense is the best defence.
Rather than the consistent protection of shields, a cloak gives you a front-loaded period of excellent defence, followed by a period of relative vulnerability while it's charging. You need to use that initial period to disable the enemy's ability to fire back at you while the cloak's recharging.
Generally speaking, you'll cloak through the enemy's first volley, but you'll be vulnerable for their second. It's therefore your goal to prevent them from getting off that second volley, by disabling their ship before it comes to that.
All about timing
In more detail, bear in mind that your weapons will charge while you're cloaked but the enemy's won't. As such, cloaking can be a bit like a western gunfight - it's all about who fires first. In the best possible situation, if you charge faster than the enemy, you can get off three "free" volleys:
- Your weapons charge before the enemy's - you take out their shields with your laser, then damage their systems with your mini-beam.
- The enemy fires his volley at you; you cloak and it misses.
- Your weapons charge back up while the enemy's are at zero.
- You fire a volley coming out of cloak.
- Both you and the enemy are now charging weapons from zero - and since you're faster, you get off your third volley first.
As such, the skill of the crewmember manning the weapons system is incredibly important, and both the Automated Reloader and Weapons Pre-Igniter augmentations are great for getting more shots off before the enemy's second volley, which you can't cloak through.
If you charge slower than the enemy, you'll normally only get one "free" volley off. The Stealth Weapons augment helps a lot here by letting you cloak on the enemy's first volley, fire shortly afterwards, and come out of cloak with more charge on your weapons.
If you are faster at charging weapons than your typical opponents, upgrading your cloak can give a huge bang for the buck. Notice that in step 4 above, you cloaked immediately after firing, and are able to fire again as the cloak wears off. If your weapon charge time is around 10 seconds, upgrading the cloak to last 10 seconds lets you get this extra "free" volley. Otherwise, it won't do enough to let you get your third volley off before the enemy's second.
Asides from that, it's mainly about the alpha strike with the stealth ship (as is hinted at by the Bird Of Prey achievement). Spend your scrap on buying more weapons, and fitting/powering them. Lean towards faster-charging weapons that you'll be able to fire in rapid volleys as above, rather than heavier weapons (5x1 damage from a Burst Laser III is awesome for a slugger, but the 19 second charge will inhibit your ability to fire volleys around stealth).
You really need to keep upgrading your weapons, in order to punch through shields and disable enemy weapons systems before they get another chance to fire.
Asteroids are bad, bad, bad. Thankfully(/essentially) you start with the Long-Range Sensors augment; don't jump to a node with environmental hazards.
Anti-ship drones are an incredible pain, too - they're almost like a portable asteroid field (regular, light damage). In this case I try to cloak immediately and prioritise the enemy's drone control bay above its weapons bay. I feel this is still a bit hit-and-hope; I don't feel confident dealing with drones in general.
Beam weapons don't have travel time like lasers and missiles, so you can't cloak while they're "in the air" to avoid all damage. I'm not sure whether cloaking when the first room is hit reduces damage to the other rooms; I suspect not as beams don't seem to ever miss.
Ships that have weapons of different sizes don't have a single volley that you can dodge (e.g. if their guns fire 3s apart, you cannot cloak while they are both in the air). This lessens your front-loaded defence, since you can't avoid all of their first shots. In this case you need to decide which weapon will hurt less, and cloak for the other one(s).