Why play Zac?
Zac is a powerful ganker and has a pretty fast clear time. He can also stay fairly healthy while in the jungle and is amoung the few junglers that can make it to level 3 and have the health for a proper gank in one clear. His damage can stay relevant throughout the game if you have multiple tanky bruisers or he can be a 1 man front line if your team is squishy. He has a great kit for forcing engagements and his passive lets him give absolutely 0 cares about diving towers. He is my go to jungler and I enjoy playing him a lot.
I personally don't like taking anything besides flash and smite, smite helps you secure objectives and clear the jungle faster and every jungler should have smite because of this. Flash helps you position your ult and get into that back line, make plays, escape death, or save allies which makes it the only other alternative in my mind. Your job is to CC and having a flash to get your ult on that backline and chase that kill can make all the difference in the world.
I talk about this some in the tooltips (mouse over the item builds) but I feel like it deserves some more in depth looks. Your item build will depend on how the game is progressing and what your team comp is like. Do they need you to be tanky and soak up all the damage or is there room for you to build offtank with more focus on damage so you can make the carries afraid of the damage you deal instead of just the CC you bring to the table.
You also need to know the enemies team comp and what kind of damage they are dealing, an enemy team with a poppy, zed, draven, and support karma for example are very heavy on the ad damage but poppy's q deals magic damage and even turns the AD damage into magic damage, and karma brings a fair amount of magic damage so you will still need some MR to stay tanky, knowing what type of damage to prepare for is essential and can mean the difference between winning and losing, I have seen games lost because the tank built all armor against a mixed damage enemy team simple because they all buy AD items. Some champions still deal a lot of magic damage despite being AD centralized so know thy enemy.
Here are some great item combos for countering enemies.
Warmogs/Spirit Visage - Syncs with your passive and is GREAT for enemy mages and poke comps in general.
Randuins/thornmail - Great against super heavy AD damage from squishy characters like Vayne and Tryndamere
Banshee's Veil - Not a combo but this item is mostly for fighting against hard engagers like Vi, Malphite, Amumu, or Sejuani, or burst mages like Veigar or Annie, anyone who relies on getting off a key ability to be useful and initiate the fight, having that spell shield means even if they engage you can counter engage the enemies backline so they have a harder time following up. I see an awful lot of people buying BV at improper times so I hope this helps. You can also buy it if the enemy is completely magic damage focused as the health and MR and spell shield are helpful. On Zac however Spirit Visage is better for the gold against a balanced team.
Zac has a pretty easy jungle path. Blue side I do golems/red and then wraiths and blue if I don't foresee a counter jungle. If they have a high mobility jungler like Lee or Shaco I might skip blue and go for a gank, or go to blue immediately and ward carefully so I can see counter jungling coming. Zac stays pretty healthy throughout his jungle clear so even if the enemy jungler does come you can usually escape easily with e or flash if need be.
The most important part of jungling is to control the objectives and smart ganks. Ward the enemies jungle so you know where they are and gank where they are not or counter-gank when they do.
The blob shuffle
While jungling, grabbing your passive blobs does more then keep you healthy, it also lowers the cooldown of w. Keep w on cooldown as much as possible and always grab the blobs, you can e into a camp and w immediately then q and grab the passive blobs from your first 2 abilities, w again, grab the q blob and w blob again and w a third time. It's a bit weird to follow but if you follow a w -> pick up 2 blobs -> w routine you can max out the damage from w without wasting blob cooldown. This is a small advantage but it matters a lot early on so practice it. If I have time I might add a video showing this better.