Sometimes the best Zerg strategy when confronting a Protoss revolves around being very mindful of what your opponent is doing and having quick reaction time.
In order to win this type of match, you must use the Zerg's characteristics that enable it to expand extremely fast.
The Zerg can quickly create a lot of cheap units. The main Zerg tactic in many fights is to swarm your opponent with loads of units. In order for this to be possible, the Zerg have been designed in such a way so as to allow them to quickly extend their presence over the map. This also works from the storyline point of view and adds to the flavor of the race.
You need to expand as early as possible because you will need the extra resources. The only way to beat Protoss units is with the correct counter-unit in the correct numbers. Beyond the choice of which Zerg unit to use in battle, it all comes down to the numbers that you have at your disposal. Of course there is a great amount of skill involved as well.
Thankfully the idea of quick expansion has been built into Starcraft 2 maps. Every spawning point has its own natural expansion point - this is usually another mineral and gas field very close-by. While these natural expansions are map features that every player and race can benefit from, it is the Zerg who claim the first mover advantage.
One very quick expansion build order has you make Drones till you reach fifteen supply. Once there, make a Spawning Pool and continue building Drones till you get to sixteen supply. At that moment you need to put down your second Hatchery and start making some Zerglings for defense.
Now mind you, the second base isn't necessarily there for immediate resource gathering. It's there for when you need extra resources later your match. However, should you need to start mining there immediately, begin with gas. The Zerg requires large amounts of vespene gas in order to create their more powerful units, such as Roaches and Hydralisks.
Defending against the Protoss
Defending your expansion and your main base is a very delicate matter with such a tactic. However, you can make great use of the Zerglings you build in the meanwhile and the two Queens that you will have to produce in order to keep your numbers up. Add a couple of Spine Crawlers and you can defend against an early Zealot rush.
Attacking the Protoss player
Even though you are expanding quickly, that should not be the only focus of your attention. Starcraft 2 is a very scouting-intensive game and you need to be aware of what your opponent is doing at all times. You can use one of your Drones to see what he is planning on doing, then using your Zerglings to test his wall. A Protoss players will usually block his ramp access with a Pylon, a Gateway and a Zealot. In some cases your Zerglings can breach it and start attacking the Pylon.
If you do manage to get your Zerglings in his base, bypass any Zealots he might have and head straight for the Probes. You can deal a fatal blow to his economy with such a move. On the other hand, if the attack isn't all that successful, you already have an expansion down and have been producing more units anyway.