Hi,
I saw all the other guides seem really different from how I play Nautilus, so I wanted to start this one.
It's really short because it's my first guide ever, but it sums up quite well what I think is the most important things in playing Nautilus.
Feel free to say what you think about my guide, every suggestion will be appreciated and considered.

And most importantly:
Have fun!

Once, Nautilus was a sailor commissioned by the Institute of War to explore the uncharted reaches of the Guardian's Sea. This expedition took him deep into unknown waters where he and his crew found a vast section of black oozing liquid that none of the crew could identify. Though their job was to investigate anything new that they found, no man aboard was willing to brave the murk except Nautilus. Only moments after he donned the hulking diver's suit and climbed over the ship's rail, something lurking in the muck grabbed hold of him. He clung to the side of the ship, but the thing below pulled him fiercely, rocking the entire ship. The other sailors grew afraid and made a terrible decision. As he stared and pled for help, they wrenched his grip free of the rail. He tumbled into the ink, grabbing the anchor in futile desperation. Dark tendrils enveloped him and he could do nothing but watch as the dimming outline of his ship faded away. Then everything went black.

When Nautilus awoke, he was something... different. The great iron suit had become a seamless shell around him, concealing whatever awful truth lay inside. All the details of his memory seemed fuzzy and indistinct but one fact remained clear: he was left here, alone in the sunless depths, to die. In his hands he still clutched the anchor that belonged to the men who had condemned him. Having no other purpose, he took this clue and trudged - too heavy to swim or run - in search of answers. He wandered without direction or sense of passing time in what felt like an eternal dream. By the time he stumbled upon the shores of Bilgewater, he could find no traces of the man he was. No house, no family, no life to which he could return. Terrified sailors who'd heard his tale directed Nautilus back to the Institute, but the summoners refused to relinquish the names of the others they commissioned. By then Nautilus had learned about the League of Legends and there he saw an opportunity to discover and punish those responsible for the time and life he lost.

''When consumed by utter darkness, there is nothing left but forward.'' -- Nautilus.

Pros :
1) Clearly a Big Daddy from Bioshock
2) Fast jungling
3) Fun and effective ganks

Cons :
1) Low damage

Seal : Armor flat seems the only viable choice, it helps you jungle very fast and will improve your tankiness throughout the game

Glyph : Same as above , Magic resist per level will improve your tankiness. Another viable choice may be cooldown reduction per level: more slow, dredge lines and ultimates!

Mark : I choose attack speed because, again, it helps you in the jungle. I would recommend them only for that. I'm gonna try magic pen soon, in order to decide if you still can jungle fast enough

Quintessence : Same as above, Armor pen for the jungle

Your major priority is to improve your jungling capabilities, so i took armor, Bladed armor and extra damage to minion.

Health is another important stat, it hardens your shield and therefore let you deal more damage, take them.

I found cooldown reduction really important in every tank so i prioritized Enlightement on Initiator, but it's to be noted that Nautilus is kind of a slow Champion, so you may want to try Initiator too, especially if you can't land Dredge Line :)

Smite and Flash are strongly recommended and actually i don't see any other spell to replace them with.

If you feel confident you can try to change Flash with Exhaust, that'll grant you the possibility to completely disable the enemy carries.

Here's a brief walkthrough to what I found to be the most effective jungling route for Nautilus.

1) First of all, take a point in your w and start at blue (ask for a pull and cover), if you stand in the blue golem path while it is being pulled, this will try to circle around. It means that he will take more damage before changing his aggro.

2) Now you can choose between the following:

a) If the mid lane seems gankable, take a point in your q and move toward mid.
You can either gank as soon as you arrive or wait for your teammate to reach level two and (likely) to learn a spell with crowd control (root, stun, slow...) - this would help you a lot in succeding the gank.
I usually gank as I arrive, because of tiem saving and my main goal here is not jet to get a kill but to force the enemy mid laner to flash/ghost away.
After the gank, move toward wolves.

b) Take a point in your e and move to the next point.

3) Take out wolves, wraiths and golems (spare your second Smite for the last ones)

4) You should have reached level 3 by know, so take a point in your missing spell - q or e, depending on your choice at point 2 - and go gank top (if your on purple team), after that (or directly if you are in the blue team) gank mid.

5) Now you have to go base, finish your Philosopher's Stone.
If one of your ganks was successful you will find you can also buy Boots of Speed- which would make your next ganks way easier.

How the rest of the game is played depends a lot on how the game is going.

I usually take red at this point an then start ganking the easiest lane - meaning the one where it has not been pushed or where someone in the enemy teamis particulary low or out of summoner's spell.

Remember, your goal is to finish the Mercury's Treads, Hearth of Gold and then Rod of Ages.
If the game is going really well, you can finish all of the above approximately at minute 20.

Blue golem should normally be given to the mid laner, but ask if he/she really needs it, because at this point you will find yourself quite mana-starved and that will reflect highly on your gank effectiveness. Thus ponder you choice very well.

Start off with Regrowth Pendant and one Health Potion.
After that, go for Philosopher's Stone, Boots of Speed, Heart of Gold, Mercury's Treads, Rod of Ages. These core items should be finished in early/mid game.

Choose your next item depending on the most powerful enemy:
ad carry -> Sunfire Cape
ap carry -> Abyssal Sceptre

After this item which gives you some defense stats as well as boosting up your damage a little, finish your Shurelya's Reverie - you will find essential both the passive (extra cooldown reduction and mana regeneration) and, above all, the Active unique.
After buying your Shurelya go for the other defensive item (Sunfire Cape or Abyssal Sceptre)

At this point, build your Heart of Gold into a Randuin's Omen for extra stats. Dont' forget to use its active bonus.

Nautilus is to be feared because of his arsenal of cc and aoe damage.
In early-mid game, where your abilities shine, you have to gank as often as you, your teammates will appreciate the large amount of kills and assists you will grant them.

In late game you'll become the main initiatior and tank:
Try to land your Dredge Line on enemy carry, and soon after auto attack them to proc your passive, your slow and eventually ulti will gran the kill.
For the whole duration of teamfight never forget to protect YOUR carries.
Your passive will allow you to block every attempt of the enemy to reach them, even if you already wasted all the other spells on enemy carries.

Remember your goal he's not to get kills (You got masteries for gold on assists, and items for gold per second), your goal is to allow your teammates to get the most from teamfight.
So if you had already stunned,slowed and so on a target, and he's likely to get killed by your team, CHANGE target, Dredge Line or use your ultimate on a another enemy and give your carries a second kill.