Basic Rundown:

Malefic Visions - The Key to a Malzahar victory

Be sure to use it on minions when they are low enough for it to kill them. If it does, it will hop to another nearby target. Basic attack minions down low enough to allow Visions to kill them so you can bounce it around an entire wave and clear it faster.

Resetting E is the key to massive damage with Malz. This is why you max Q second in my build. Call of the Void resets the duration of Visions, doubling it if you time it well. Because of this, you'll be using Q as a main portion of your combo, you might as well upgrade it second for more damage and a longer silence. Your Ult also resets the duration, so use these in conjunction to effectively triple your damage from Visions. Plus, Voidlings focus Vision infected targets, so while they're weak, still stuff to bother the enemy and importantly, block skillshots.

Nether Grasp - Use it intelligently

Your Ult can ruin someone's day, but can also ruin your's if you're not smart with it. Simply put, use it AFTER your E, and typically your Q too. This will mean you've effectively silenced your opponent for 4.5 seconds if timed well, and in that time done a huge burst of damage, and potentially reset the duration of E again to add even more. If your Voidlings are there too and the enemy is retreating, they can usually finish them off.

Using your Ult to engage is a horrible idea, you won't get nearly enough damage out to kill an opponent and you'll leave yourself very vulnerable. Also, using your Q early will put it off CD in time to use again at the end to finish a fleeing target off.

Laning

Basically, just make sure to use E and basics to get as much CS as possible, and Q to poke your opponent/weaken the ranged minions so your E can hop to them and finish em to reset on a new minion and keep chaining.

You don't want to fight early. Focus on farm, Lost Chapter will help you with mana sustain during the early game. If your opponent has lousy lane clear, punish them with as much pressure as you can by cycling to get 3 Voidlings.

Voidlings - The Ultimate Defense

If you're slick, you can block a skillshot by dropping a Voidling. Don't save them for this unless you know the situation is about to arise, just keep it in mind. Your Voidlings are also excellent wave clear even when not maxed, especially in conjunction with your E if you can get the E to bounce around.