Alright so I wrote this guide day 1 and have played a lot more Viego since. Yes, I am a Gold player and these other people are apparently all GrandMasters, but I hope you will give this build a legitimate try and get back to me. Anyways, I've changed about 50% of my guide but I'm gonna keep it short and sweet this time. To give a reference of how I've been doing with Viego:
Gold4 Ranked:

13+ game win streak currently:

Clearly this is anecdotal, but I have to be doing something right.

Alright the new build is simple and straightforward.
Step 1) 5 camp clear, contest scuttle at 3:15 because of Viego's decent dueling. Look to full clear again and recall
FIRST BUY:

Blade of the Ruined King

and either Berserker's Greaves or Mercury's Treads or Plated Steelcaps depending on their team
NEXT BUY: Kraken Slayer
THESE TWO ITEMS ARE ENOUGH DAMAGE FOR THE GAME, YOU WILL BLOW UP BACKLINES AND SHRED FRONTLINES
From this point forward you build bruiser items, most importantly Sterak's Gage and Death's Dance

The more and more I play Viego the more I feel his power is in the early and late game. He power spikes somewhat at 1st item, a lot at 2 items, kind of declines at 3, and then hyper carries at 4. You should look to end at 2 items or at 4.

So What's the Gameplan with this Build?

Simple: Everything. Blade of the Ruined King + Kraken Slayer synergize great with his combo and 2-hit passive, allowing you to pump out 300-500 true damage and 55%+ current health damage in less than 2 seconds. You can use this to get good picks using your Harrowed Path or use it during teamfights to shred frontline tanks. Alternatively, you can use you Harrowed Path to get a flank during teamfights and melt their backline.
The thing about this build is that you aren't building full damage and you aren't building that garbage Goredrinker build that's just gonna get turbo nerfed next patch. Instead you're playing Viego how he was meant to be played: building hybrid so that you can kill THEIR tank and become tanky or kill THEIR ADC and become a damage carry. It's simple, and it's effective.

Alright everything below this is still mostly reliable from my day1 analysis.

So lets hit the basics on HOW TO CARRY ON VIEGO:

-Early Game-
+ Go for a 5/6 camp clear and contest river scuttle, after dueling on it you can potentially double rift scuttle and get early lead
+ Viego has very strong early game and jungle clear, if you have bot and mid prio and can get a gank going, Viego can solo the dragon at level 5 and can get a free early drag (BE CAREFUL, IF YOU ARE CAUGHT AND YOUR LANE CANT ROAM YOU COULD THROW YOUR LEAD)
+ Typically go for 2x full clear on jungle before recalling, recalling at around 1500g

-Mid Game-
+ Try and get pressure side laning, your wave clear is decent and if they send one to duel you you can usually use bushes + your stun to win most fights
+ Play for RIFTS AND DRAGS HARD!!!!!! Viego does %current health damage to objectives, which makes his neutrals fast and healthy
+ Look for flanks on team fights OR picks where their tanks are caught out. Once viego has BORK + his Q passive he can shred most tanks who only have 1 item. Alternatively, if Viego can get on the back line and use his ult resets you can carry teamfights.

-Late Game-
+ Play around your team, you are a bruiser with 3-4 items and need to be able to pick out their front line with your full combo, using ult to dive onto the backline can be good but IF THEIR BACKLINE HAS MOBILITY YOU ARE GOING TO LOSE IF YOU DIVE. People like Lucian, Karma, and even Ziggs are very hard for you to pin down if you don't kill them in one combo.
+ His split pushing is once again not awful, but he is certainly incentivized to look for fights around his team.

If on the other side I like to get the bot leash and start blue, ultimately taking these camps again. This gets Viego to the scuttle at 3:10, which gets him in a good spot to take it or duel on it, which is pretty good for Viego.