This is the first build I've made for LoL, and Nasus is the first champion I've spent time working with and really building. It's still a work-in-progress, so take it with a grain of salt.

I started off playing Nasus with a great build called 'QQ Cuz of My Q' by Asterul, and I definitely recommend it to anyone trying Nasus for the first time. My build does borrow heavily from that one as far as runes and masteries, but I've changed it up and made Nasus slightly more DPS-oriented than tanky here. Don't worry, though; he's still made for tanking/soloing.

Pros:

Properly farmed Q creates crazy damage output
Tankiness
Has a great drain with his E
Has a great slow (W)
Pretty awesome against other champs in 1v1 fighting
Possibly the greatest voice in the entire game
Walks like an Egyptian :P

Cons:

Can be very slow to start building strength
Not much in the way of CC
Q is reliant on proper farming/lane placement
Not quite as useful in teamfights as other tanks can be

Greater Mark of Desolation: this increases your damage output exponentially. A must for this build.
Greater Seal of Replenishment: Ups your mana regen. Nasus' early-game mana pool is not great, so you'll definitely want these.
Greater Glyph of Focus: Increases your cooldown reduction (CDR). Nasus' Q is your best weapon, and this will help make it faster to use.
Greater Quintessence of Desolation: Like the Mark of Desolation, this will increase your damage output. I only use two of these so I can add:
Greater Quintessence of Clarity: Another mana regen rune. With this, your mana pool gets a little better, and will regenerate faster.

My masteries for Nasus go 9/21/0.

Defensive masteries let you stay in the lane longer early-game, which lets you farm the heck out of your Q. As for the offense,

The thing to remember about Nasus is that his Q attack, Siphoning Strike, "permanently gains 3 damage whenever it kills an enemy." The best time to build this damage is during the laning phase.

Farm. Farm like your life depends on it; if you don't your late-game damage will be extremely poor. If you're having trouble last-hitting with Q, lay down a Spirit Fire (E), then pick off the minions caught in it with Q.

Let's talk items for a second. There are a couple of viable paths you can go, but my personal preference is to start out with the Sapphire Crystal and a Mana Potion. Early game, Nasus' mana pool is very small, and his Siphoning Strike makes him a very mana-hungry champ.

Once you've spent some time farming, you should have enough gold to pick up our next two items in one fell swoop. Ionian Boots of Lucidity will up Nasus' movement speed and lower his CDR, while Sheen is the quintessential Nasus item.

From there, you want to start building Stark's Fervor. Combined with Nasus' passive ability, at level 11 you'll have a 40% lifesteal! Who doesn't love being able to stay in the lane for ages while enemy champs have to keep running away?

These are the real core items of the build, but if you've been farming diligently you should have enough gold to keep getting things. I like Shurelya's Reverie because it raises your health, gives you extra mana regen, and ups your CDR by another 15%. Morello's Evil Tome is also a great source of mana regen, AP, and another 20% CDR.

As far as Summoner Spells go, I find that Ghost and Ignite are the best combination for this build.
Ignite is a great spell because it's surprisingly versatile. Early game, I like to use it to scare off harassing enemy champs; late-game it's a good way to ensure a kill on a weakened enemy. On champions like Sona, Soraka, and Kayle, its heal-dulling effects are priceless.
Ghost is great for escaping from fights and retreating when need be, but you can also use it to dart in, smack an enemy around, and dart out again. I do not recommend the latter until late in the game, however.

As far as Nasus' skills go, you want to start off strong with Siphoning Strike (Q) at level one. Farming is essential here, so when you hit level 2 (and you'll do that fairly quickly), get Spirit Fire (E) to help you corral large groups of minions and drain enemy champs.

Alternate them over levels 3 and 4, and pick up Wither (W) at 5. At 6, you can finally get Master of the Sands (ult).

From there, it's free-form. I like to level up Q and E alternating, so they're maxed out at roughly the same time. Get your ult leveled when you can, and grab W when you're done.

I like to solo either mid or top, depending on which team I'm on (mid for Blue, top for Purple). Nasus is built to solo, and it helps if you're the only one competing for minions. You may have some trouble with certain ranged champs (Ezreal, for one), so don't be afraid to ask a teammate to swap if it's really bad. Better to switch than feed.

Your Q actually does a lot of damage to turrets, so as the game edges later, be willing to take a swipe or two at a turret if you can.

Make sure you're last-hitting with Q. As I stated in the 'Farming' section, if you're having trouble, lay down an E, then pick off the caught minions with Q.

With a pumped-up Siphoning Strike, you will be a master of control in the endgame.

I recommend going for the main support in teamfights, and hitting them with everything you have. Throw this sequence: W > E > Q > Ignite, then back out for a cooldown. This will leave them weakened and slowed, which makes them an easy target for your team.

In short, Nasus is a hell of a champion. Tankiness, damage, and a voice like buttah. What more do you need?

I'll be updating this guide as I tweak it, and any comments and criticism would be welcomed!

Thanks for reading! Hope to see you on the field!