What's good whippers and snappers. Emo here, with my covered-in-seaweed friend here.
Nauti
"Yeah, keep that anchor away from me, Dr. Gillyweed."
Nauti
Anyways, I'll be covering one of the most powerful champions in the League. I tend to give many champs that title, but this bad boy has been overthrown with nerfs ever since his release - no buffs at all.
Nauti

Nautilus, the Titan of the Depths

I bought him, played a game, failed, and dunked his a$$ in the ocean. Then for some reason, I decided to go 'unconventional' for once, and found out that ColbyCheeZe was not lying: Ability Power Nautilus packs quite a punch. A punch with an anchor that is, so inb4 pain. Anyways, it's hard to see something outside the meta in mid, or in any lane for that matter. You won't see a Taric on the solotop lane, even though it works pretty well. Same goes for Nautilus. You see him without Smite, it's an insta-report for trolling.
Nauti
Believe me when I say that Ability Power Nautilus is oh so damned strong. However, seeing as how your damage is on a short delay, and true tankiness is not achieved in this specific build, it is NOT advisable to initiate. Ever. Sure, you could thrown down a Depth Charge and let the real tank take it from there, but you don't want to soak the damage. When you do, you'll get merped since you're still the AP carry, and then it's a 5v4.

So let's quickly align pros and cons so that you too may wreck ships with Nautilus, or decide not to.
Nauti

+ Strong bursting capabilities
+ CC King, now from the mid lane
+ Devastating hook
+ Tankiness increasing by level
+ Pretty damn strong
Nauti

- Terrible dueling power w/o Blue
- Melee mid is melee, hard farm
- Easily outrangeed by champs
- Anchor won't save you all the time
- So. Damn. Slow. Ohmygod.

Every first attack Nautilus lands on an non-allied unit deals significant bonus physical damage and briefly roots the target in place. This has a 12 second cooldown on each individual target, unaffected by CDR.
Nauti
It's what made Nautilus a really strong ganker, as it gives your allies a short moment of time to whip out ye olde CC and make the gank official. During the root, the enemy can't Flash, which spells certain doom for those who have slow reaction times. It also gives you time, in your own spell rotation, to successfully use Riptide and Titan's Wrath on the enemy without wasting them because Nautilus is so damned slow.
NautiNauti Nautilus hurls his anchor in target direction. If it collides with an enemy unit, him and the unit are pulled together, while the enemy is dealt magic damage. If the anchor collides with terrain - Turrets, Inhibitors or solid walls and trees - Nautilus pulls himself all the way to his anchor, and the cooldown is halved.
Nauti
This is, obviously, your primary gap closer. It's a magic damage nuke, which is nice to have in a standard spell rotation. Phreak said:

"If you're aiming for an enemy near a wall, you're guaranteed to pull yourself forward, even if you miss."

He's right, and you should put it to your advantage. It's your most effective gap closer, and often times it'll pull out a Flash or provide the enemy a closer look into the aspects of a near-death experience.
NautiNauti Nautilus places a damage-blocking shield on himself, scaling from his bonus health. In addition, for as long as the shield remains, his basic attacks apply a meh damage over time on enemy units in front of him, dealing magic damage over 2 seconds. Casting this ability resets his auto attack timer and gives him a vicious swing on his basic attacks.
Nauti
Jungle Nautilus loves this ability. I think it's meh at best. With Nautilus being so damned slow, it's a godsent if you get... two basic attacks off without the Blessing of the Lizard Elder or a Rylai's. You'll have to pop ult if you want more than two. I rank it last, because it's just not good enough to rank early.
NautiNauti Nautilus sends out a shockwave around him in the forme of three exploding rings, expanding outwards. Each explosion deals magic damage and slows affected enemies. Enemies take 50% reduced base damage from additional explosions, but the slow duration is refreshed.
Nauti
Without a doubt, Nautilus' strongest ability, and one that'll wreck sh*t if overlooked. After Lv.7, it'll one-shot caster minions with the second explosion, and one basic attack will kill the melee minions at that point. Additionally, it's a really powerful ability in terms of sheer damage. Two explosions = 330 damage @ Rk.5. Max as soon as possible. Best damaging ability, best farming ability, and amazingly strong when paired with Dredge Line, and Titan's Wrath to some extent.
NautiNauti Nautilus marks a single enemy champion. From that moment, a shockwave will chase the target, dealing magic damage to and knocking up all enemy units it passes through. When the shockwave reaches its target, it will deal double the damage and knock the target up, after which they are stunned.
Nauti
You'll be amazed as to how far the initial cast target can be away for it to still be successful. Just have vision and pop it over a wall; they'll never see you coming. Reeling this through your basic spell rotation is really strong, as the base damage at Rk.1 is amazing, especially paired with a good AP Ratio and your other abilities. Damage increases, Cooldown drops: rank whenever you can.
NautiNauti
Obviously, I rank Riptide as soon as possible. It's not that it carries a big range or anything, but it's simply one of the strongest farming tools in the game, able to clear caster minions in an instant and take chunks of the enemy's HP in the same time. Next goes Dredge Line. Without ranking, its base damage is sh*t on a stick, and its cooldown is rediculous at the same time. I rank Titan's Wrath last. I know some will disagree, but it's relatively simple: Nautilus is too slow and does not carry the innate bulkiness to rush a Rylai's straight off the bat, which you'll need to get more than two hits off. At the same time, it's extremely cost-inefficient, and the shield won't persist for nearly half as long as you'd like. For obvious reasons, I take Depth Charge whenever I can.

Nauti
My runes are my standard caster runes, with Magic Penetration, Flat Mana Regeneration, Ability Power per Lv., and Flat Ability Power. These runes provide stable early game damage, well-scaling damage as we won't go full out RUSH DEATHCAP AND ARCHANGEL'S, but with a bit of tankiness in between, and some Seals to keep up with Nautilus' unforgiving mana costs.
NautiNauti
On my Masteries, I go for the same standard caster page. I take everything Ability and Ability Power related in the Offense Tree, as well as Summoner's Wrath and Butcher for extra damage and better last hitting, respectively. In Utility, I take Increased Mana and Movespeed, because I like my abilities, but I hate being slow. Additionally, I take Runic Affinity , because Nautilus becomes a godlike killing machine when he carries the blue buff. And what's better than half a minute of extra killing machine ^^

This is merely a quick overview of the items I like buying on AP Nautilus. You don't have to agree with the items I buy, or the order in which I do, but at least read through it and know WHY I do. It all makes more sense then.

Boots:: I've said this multiple times, and I'll say it again: Nautilus is very very slow. Not only do we take the Movespeed mastery in Utility, we also take Boots to go. These are by far the best starting items, but they also make up for what he desperately lacks.

Doran's Ring:: Not a big fan of these, but buying one or two of them can really help you in lane. Not only damage-wise, but sustaining too. The 16 bonus health on Titan's Wrath is negligeable, but the 2MP/s is really strong, especially when paired with 30AP.

Mercury's Treads or Mobility Boots:: It depends on which role you want Nautilus to fulfill. With him having godlike amounts of CC, he could be played as a really strong ganking midlaner, using the movespeed to get around. On the other hand, his farming capabilities are insane too, which makes going Mercury's Treads and staying in lane another viable option.

Abyssal Mask:: I take this guy over a rushed Rod of Ages because of the underlying maths and what it actually brings me over the Rod of Ages. This item gives me effective health vs. all magic damage dealing enemies, rather than true health against the first enemy that's strong enough to take it off. Simple matter of math. Additionally, the AP is nice, and the MR reducing aura is a gank-killer, but makes ganks really strong if executed properly.

Rabadon's Deathcap:: You won't be able to instagib the enemy mage with it, but you'll get close to being able to do so on the enemy AD Carry, providing they don't build defenses on you. Best mage item, hands down. It vastly increases your damage output, which is awesome, obviously. And even though it doesn't make you any tankier, it does make you quite the damn bit scarier.

Athene's Unholy Grail:: If Rabadon's Deathcap is the best item, the this guy is a close second. It provides all the stats a mid-laning mage wants. MR for bulk; AP for damage; MP5 for spam; CDR for even more spam, and a Mana Regenerating passive if you do well. Not to mention the stats it provides are simply off the f*cking charts. You don't mess with the Athene's.

Rylai's Crystal Scepter:: Even though it's rediculously underbalanced compaired to a Frozen Mallet, it's still a really strong item. When it'd sometimes be hard to follow up with a basic attack after Dredge Line'ing an enemy, this will keep them in range for a bit, making that just a bit easier. Not to mention it practically becomes a Frozen Mallet with Titan's Wrath activated thanks to the DoT, and one hell of a slow with Riptide. Just a good item overall.

Frozen Heart:: This'll miss the CDR cap by 1%, but we'll be pretty damned tanky, it's too good to leave behind. I take this guy purely for the CDR and Armor it provides. I hate being shredded by the AD Carry before I can put myself to good use, and this allows me to put my 600~ AP to some use before dying, if I die at all, that is. It'd been better paired with an Archangel's Staff, but hell. Still good enough.

I'll provide a short but sweet overview of some stats this build will provide.

At Lv.18 with full build but without Elixirs, a botgame yielded me about 600~AP. This translates into rediculously strong burst damage, even if Titan's Wrath won't persist. The amount of stuns Nautilus brings to the table allows for Titan's Wrath to stay up longer than the enemy'd like, though, but I just go for the lower numbers.

In addition, a full item build grants both 170~ Armor and 170~ MR. With the League-calculations, this turns into a 63% damage reduction. Also, thanks to both the Abyssal Mask and the Frozen Heart, we have a few nifty auras to deal increased damage, and reduce the effectiveness of enemy gunfire, respectively.

Next, we have a grand total of 39% CDR with the Athene's Unholy Grail, the Frozen Heart and the Sorcery mastery. This drops the cooldown of Depth Charge to roughly 49 seconds, which is pretty damn strong. It also drops Riptide to 6.1 seconds, which is extremely nice to have in conjunction with the same cooldown on Dredge Line. The slow from Rylai's Crystal Scepter will keep enemies near, and by the time they are freed from all the stuns and slows they faced, both Dredge Line and Riptide will be up again.

Nautilus is an intricate champion, very very different from most AP Carries. You can name me any mage, there's either a skillshot or AoE skill on a decent range. Nautilus does not have that. As such, you're going to need to stay really passive in lane until about Lv.3 or Lv.4, when you have access to all your skills. A basic spell rotation should be something like this:

Dredge Line --> Staggering Blow --> Riptide --> As many hits of Titan's Wrath as possible.

This is guaranteed to take off at least 50% of their health bar. Keep in mind that Nautilus scales extremely well, both with innate tankiness - although that one never really takes off - and with his damage figures. The later during the laning phase you're going to get, the more damage you will deal. It's as simple as that.

During later stages of the game, when you have access to more powerful and sustained versions of your abilities, you should look for ganks. The AD Carry is the guy/girl you'll mainly want to focus down as soon as possible, as she's the main threat to your team if left alone. In the fortunate case of you finding them on their own, here's how you'll want to get to work:

Dredge Line -> Staggering Blow -> Titan's Wrath -> Riptide -> Depth Charge -> Riptide

If you have it available, an early Ignite will amplify all your damage slightly, as well as reduce the effectiveness of heals they probably will receive during the 8 second fight.

What you should keep in mind though, is that everything Nautilus has - albeit very powerful - is either a skillshot or a point-blank AoE skill. You can't just walk up to an enemy, pound them with single target nukes and walk off. You'll get hurt if you try to. Any and all damage you'll do will have to involve Dredge Line, since it's the only way to get close to the enemy. If they are stupid enough to allow you to walk into melee range, you can't lose your lane.

A quick overview of a full spell rotation at Full Build:

Dredge Line:: 240 + 450 = 690 Magic damage
Titan's Wrath:: 260 + 480 = 740 Magic damage (in the case of 2 hits)
Riptide:: 330 + 450 = 780 Magic damage (in the case of 2 explosions)
Depth Charge:: 450 + 480 = 930 Magic damage

Base damage calculations turn to 3140 hitpoints worth of magic damage. Providing they buy no additional magic resist (for whatever dumba$$ reason), and taking the Abyssal Mask-aura into account, they'll have roughly 10MR. This means they take 2855 points worth of magic damage through a full spell rotation. Damage enough for ya?

A quick overview of some of the enemies I've faced in the mid lane. With Nautilus being such an intricate champion, I judge things differently, as Nautilus is... 'different', to say the least. He requires a different playstyle than... let's say Gragas, and as such, wages up to mages differently than him.
Nauti Kennen's a mean bastard. He's so slippery and fast, it's hard to really fight him. Paired with him being ranged, and us having an amazing 15Armor when we go into lane, it's a hard time during earlier Lv's. Especially if he puts the passive proc on Electrical Surge to good use. However, our burst damage is WAY superior to his, even though he has a short-duration stun. We have 3 not-so-short duration stuns and just superior damage figures.
Nauti Ryze is another type of champion. He specializes in extreme single target burst damage, but has f*ckall for range, and even less for gapclosers. We have superior damage, even better single target burst (as showcased above), and we DO have a gapcloser. It's tricky, however, because Ryze can keep throwing Overloads at us, and there's nothing we can do about it. Just force him out of position and then kill him.
Nauti Twitch has to be one of, if not just the, worst mid pick ever. Not only does he seriously lack some damage early on, he is more fragile than a vase. A single spell rotation of Q+E will set him back more than half his health bar post-3, and with Titan's Wrath available, it's GG for him. Get fed off of him, then proceed to gank other lanes.

My conclusion is very simple: Nautilus is very strong, but is intricate. I don't like to jungle him, as I find his jungling damage rather lackluster - even though it just isn't. From the midlane, I adore playing him. Enemies just never see him, they won't know what to do against him. First reaction will always be 'SAVE YOUR SOULS', which is just a bad decision as Nautilus can't duel worth of sh*t.
Nauti
Regardless, Nautilus is a really strong and explosively ganking midlaner. He's expensive, but totally worth it.
If you think this guide is adequate, please leave an upvote for me, helps me keep track and stuff.
Have fun anchoring domes.
Nauti
~Emo