The biggest problem everyone has with Cassiopeia, as far as I can tell, is that she dies way too easily (aka squishy) especially since to use her best damage output, you have to stay close to the enemy (high sustained damage, low burst). This build is about giving the strategy not to enhance what is already great about this sassy snake, but to improve upon her weak points in order to counter the problems that hold her back.

In order to do this, first look at Cass' pros and cons. Note that she is not typical of most straight mages when choosing stats.

Pros:
Sustained Damage
Chasing
Mana Efficient
Game-Changing Ult
CC Slow, Self Speed Buff
Very Low Cooldowns

Cons:
Squishy as all Hell
Prone to Ganks
Prone to Traps
Low Burst (except with ult)
Very Difficult to Master
Low Map-Awareness (with this guide)

Now let's see how we can work with this.

Some people have the idea in their head that if it's an AP Carry, then we must throw giant amounts of AP at it, and everything will be fine. Not so, my friend. In fact, playing Cass has caused me to buy RoA almost invariably with all of my casters. Consider:

Her AP ratio isn't so amazing. True, she needs AP or she won't do damage.
MPen is often more useful to her given that she is a sustained damage dealer, thus she has the potential to take down tanks better than a burst mage or assassin.
Because her damage is almost all sustained, her squishiness is a big issue, forcing you to remain near an enemy.
Obviously, movement is desirable because of her proneness to ganks and her inability to make it out easily without ulting.

Thus, I have come up with priorities:

1. Movement (Boots of Swiftness, Force of Nature) allows her to harass effectively, moving in and out and dodging skillshots as well as maximizing her potential to chase and deal high DoT
2. HP (RoA, Rylai, Banshee) best thing a caster can do to counter squishiness. Also aids her in tower diving like you wouldn't believe.
3. HReg (Regrowth Pendant) this is more for early game. It's just lucky it builds into FoN. Cass has incredible staying power as long as you don't get into too much harass, but this item will let you take some harass and hopefully outharass your opponent. Since mana and AP aren't as much of an issue in her laning phase, I definitely suggest this early.
4. AP it sounds like I'm contradicting myself, but no mage can kill anything without good AP. However, note that the build prefers getting hybrid items for Cass instead of a Rabadon. If you're dominating, sure, go for a mejai, but take hybrids if you want a higher chance of success.
5. Mana (RoA, Banshee) That's really not a lot of mana. Honestly, she DOESN'T NEED A LOT. her passive helps more than people give it credit. Without burning your mana carelessly and needlessly, she almost NEVER runs out. Of course, a tiny bit that comes from these hybrids can help you turn that almost never into a flat never. Finally, her mana regeneration comes from the seals, and that's all you'll need. More is not necessary. She'll have so much staying power not even Morgana could outstay her. And you're not even using clarity!
6. MPen It's low on the priority list just because you won't need it until late when people start getting resistance. Definitely go for it, it's important.
7. Resistance More important than defense because Cass tends to have less trouble holding her own in close quarters than from a distance.
8. Defense not even in the build, but it's useful if you want it.

Don't get any other stat. No AD, no AS. That's trolling.

Runes: The runes are pretty standard Mage runes. Changing it up, getting movement quints etc won't hurt, but these runes do work. The MPen and Mana Reg are almost irreplaceable though.

Very simply, early game playstyle, particularly in mid, is "At a distance". Wait for your basic combo of q and e, and execute it when you see a window. The best thing is to be unpredictable. Don't run up to the enemy, let them come to you and test your range. If you hit them with q, your speed buff will allow you to get at least two fangs in.

Mid game, the style gets to be right between careful and bold. Don't be afraid to chase low hp, don't be afraid to help your partner, but you should still savor the staying power of your character.

Late game is fun. You can farm entire minion waves for almost no mana cost now, and your CDs will be out before an enemy can come to interrupt you. Do it! This is where the playstyle becomes "Up close and personal". Get in there and chase the hell out of your foe. Your HP should save you unless they're fed or in a group. In team fights, look for opportunities to stone 3 or 4 in the faec. 2 seconds is A LOT when nobody is moving, and hopefully a few of them are standing in your miasma.

Flash and Exhaust

Honestly, other than flash, it's hard to pick a suitable spell for Cass. Exhaust is useful, however, in positioning your q if your w missed. Or for escape from that AD if your flash was burned. With flash, it becomes interesting. In the middle lane, it becomes incredibly necessary. Your opponent may have low health from your harass and is now staying just out of your range, moving back whenever you so much as breathe. You can, with flash, go in, stun them (or slow if their reaction is good enough), and pick them off as they try to get away.

With combos like the ones you get from Cass, that just keep coming and coming, a few extra seconds of hp will mean a hell of a lot of extra damage on your enemies. When you die (if you die), make sure you take two with you or at least damage 4 of them halfway in a team fight. If you can't do that much, you just need more practice. Learning Cass takes patience. She's incredibly difficult to play right, but rewarding once you have her down.