Cassiopeia is one of the meanest and yet under-appreciated Caster in the game. Usually, when you see your ally pick Cassie, you groan at the likely-hood of a feeder. When the other team has one, you are sure it will be an easy game. Considering that Cassiopeia is my absolute favorite champion to play as, it makes me sad that so few people admire the vicious ferocity that comes with a good Cassiopeia.
Cassiopeia is not only one of the hardest champions (both to play and to spell the name correctly on your first try) but is also the best spammer, and has the most sustained damage out of any caster (She's practically an AD carry who uses AP instead).

My personal preference for playing as Cassiopeia is when I have a jungler on my team (preferably a friend of mine) because if played correctly, she can solo 1v2 incredibly well. The biggest weakness here though, is the same with most champions 1v2. If your jungler won't gank, you will generally have a hard time.
If I have to, I can mid with Cassie, but as she is very squishy, low range, and has little damage output early game, she has a hard time against long-range heroes such as Caitlyn, Ashe, Lux etc.

If your team is in need of a caster with lots of harass and sustained damage it is the perfect time to play cassie. If you are going mid, be careful of long ranged champions (as noted above). However, if you are 1v1 against shorter ranged champions (such as Mordekaiser, Xin Zhao, etc.) you can deal with the quite easily once you hit level 6 and get Petrifying Gaze, so just play defensive until then.

Sorcerer's Shoes

-1st finished item- Easy choice. 20 magic pen along with your runes means you grab these and you are ignoring all of the opponent's MR unless they have items/ masteries that give some (You ignore 15%/30 MR with these boots).

Will of the Ancients

-3rd finished item- Some people might say that because spell vamp is largely reduced with multi target spells that this is a bad item choice.
*Fact: Cassiopeia has only 1 single target spell.
*Fact: Once you grab Tear of the Goddess, you have, in essence, 2 attacks per second with your Twin Fang (and really nice dmg on it too). Now what champion with only an auto attack (2x a sec) and half decent damage WOULDN'T grab lifesteal????? None. (and your damage with Twin Fang is far more than decent)
*Fact: With Will of the Ancients Cassiopeia can stay in lane longer than most, and harass harder than most as well. You can save Flash for an enemy combo, and follow up your escape with your typical combo (which will either kill or push out of lane), leaving you at good health. Healing off of creeps with Twin Fang will give mana (through Tear of the Goddess) and health (through Will of the Ancients)

Rylai's Crystal Scepter

-2nd finished item- Simple. With this you get AP (more damage), HP (more burst survivability, which cassie needs), and a slow accompanying each and every one of your spells (Playing cassie right, you should average about 2.5 spells per second during any fight beginning once you have tear of the goddess).

Archangel's Staff

-5th finished item- As Cassie, you should average about 2.5 spells cast every second (yes, that is higher than anyone else can manage). This will fill your Tear of the Goddess incredibly fast. As a result, you have an insane amount of spammability, and mana to back it up (With a mana pool of about 3,000 and 50% mana cost on all of your spells after you hit 5 stacks on your innate). With Archangel's, you get a ridiculous amount of both mana and AP for a relatively low cost (about 2800)

Rabadon's Deathcap

-6th finished item- This simply gives AP, and is really expensive so it should be the last item you finish. (It gives you no additional stats of any kind so almost everything else takes precedence) This item is nice, but it is best used as a capstone to your build, not the basis.

The 4th item you build depends on who is on the other team, but overall, the choice is simple

Abyssal Mask

-Possible 4th finished item- This gives you some AP, 20 Mpen, and a good amount of MR. Pretty much, if you need the MR, this is the item you want. You can get the Magic resist without hurting the rest of your stats in the process. (gives no hp or mana, but MR is better than Hp, and -20 magic pen makes up for less Ap from Rod of Ages)

Rod of Ages

-Possible 4th finished item- Overall, this is a better item to get than abyssal scepter unless you are fighting burst mages (such as LeBlanc or Brand) or just heavy hitting casters (such as Veigar). This gives you hp (for burst survivability), mana (for longer sprees of nonstop casting) and Ap (80 straight up and another ~24 through Archangel's Staff) This is just a better all around item, and if you are going to get it, try moving it up the list a little so that it can fill up quickly.

Now, you may ask: "why do you have 2 sections on buying items?" because I want you to know WHY to buy the items I suggest, and then WHEN to buy items for best use and effectiveness.

Beginning Game:

When I leave base, I have a Meki Pendant and 2 Health Potion. This gives me what I need to stay in lane until I can grab the first, and most important early item in the build: Tear of the Goddess. This will give you good mana and regen from square 1, and will slowly become even better. After you have Tear of the Goddess, you want to go for Boots and then head right into Hextech Revolver. Between these 3 items, you have enough mobility to get around (and chasing people down, between the speed up from Noxious Blast and the slow from Miasma), as well as mana gain and health gain.

Mid Game:

Now, you pretty much want to go straight for Rylai's Crystal Scepter (grabbing the recipe for Sorcerer's Shoes when you are ready) so that people will stop getting away from you ( Noxious Blast will now slow them down and speed you up =D). At this point, you should decide on your optional item. If you are going with Rod of Ages, now is the time to build that (just grab it ASAP). If you are one of those weird people who likes to have catalyst of the protector before hand, you can grab that earlier, but I feel it just isn't needed, and slows down the build (just my opinion, do whatever you want).

Late Game

After that is finished, you want to finish all of your items. I usually do so in the following order:
Sorcerer's Shoes: 750 gold (If you didn't finish it yet)
Will of the Ancients: 900 gold for 50 Ap 5% spell vamp
Archangel's Staff: 1860 gold for 135 Ap, 14 mana/5
Rabadon's Deathcap: 3600 gold for 300 Ap (this includes the bonus AP)

All in all, Laning with Cassiopeia is just all around fun and entertaining.
Although I prefer to 1v2 with her, I will outline basic early game strategies for laning alone and with someone else.

Laning 1v2

This is where the fun is. Going 1v2 with such a squishy caster can be very dangerous, and sometimes dependent on allied ganks, but even when it is it is very rewarding and can also lead to you becoming very fed even early game. It's all in how you play the game.

Noxious Blast and Twin Fang are your bread and butter from when the first creep spawns until it shows the victory screen. In the beginning, you really need to be cautious. If you can stop yourself from dying or needing to recall until you can grab Tear of the Goddess, (605 gold after you have Meki Pendant). At this point, it is important to save Flash for defense. If you can guarantee a kill with it, go for it, but remember you won't have anyone to hold the fort if you die; It may be best to let a kill slip away in order to stay in lane. Harass primarily with Noxious Blast, and if they are in range, a few casts of Twin Fang. For the most part, you just want to keep them at low hp. Hiding in the bushes and waiting for them to get close can allow you to use Noxious Blast and use the movement speed to get out of range of return fire.

Once you have your Petrifying Gaze, get your finger on Flash and get ready to move. Once they are low enough hp, or have pushed near to your tower, you want to flash right in front of them and throw Petrifying Gaze immediately. Follow up with a Noxious Blast right in the middle of them (or on top of them if there is only one enemy). Within less than a second you need miasma right behind them, and then you should start letting Twin Fang fly. Whenever Noxious Blast is ready, throw it down. Make sure to target the squishiest enemy first. Done right, you can get easy double kills like this. If your jungler is able, have him set up in the bush behind them.

Laning 2v2

Laning 2v2 is alot safer than 1v2, but for the most part, the harassment is the only difference. You want to make sure you lead with miasma so that you and your ally can get up in time to hit them. Due to your nice harassment, you and your lane-mate should be able to get early kills. Your ult will only magnify the amount of damage output, adding a stun for larger damage output.

Killing combo is the same once you end the laning phase and begin ganking / team fights once the fight has begun, Flash in, and cast Petrifying Gaze. Throw down Miasma on as any of them as possible and follow up with a Noxious Blast next to your chosen target. Focus the largest threat first (carry or burst caster) and keep the other team under constant cc from Miasma and Noxious Blast

General Fighting Once You Have Rylai's Crystal Scepter

(Let's assume you are level 11 for this just for the use of accurate examples.)
Noxious Blast hits for nice dmg, but more importantly lets you get a real chase going on with someone. Your target gets poisoned, (and so now should have Twin Fang flying) and is slowed 15% until your first Twin Fang hits, (then it is 35%) and you are sped up 20%. Can they get away? No. But because you are mean, now you throw Miasma on the ground in front of them. They either take a detour, taking even more hits from you or run through it, taking even more hits from you. By now (we are about 3 seconds into the fight) Noxious Blast is back, so you lead them a little and hit them again (more slow, and you get more speed). Follow up with continual twin fangs, and noxious blasts until you get the kill. The most important thing is just to hit with your Noxious Blast. This can guarantee almost any kill.

Team Fighting

Generally, you want to just harass with Noxious Blast until the fighting begins, don't use Twin Fang unless the opponent is poisoned, because if the team fight starts suddenly while Twin Fang is suffering from it's 5 second cooldown, you (and your team) will be at a great loss for damage for the first few (most important) seconds of the fight.
Once the fight begins, you want to throw Miasma on top of the most mobile of the enemy team (the melee champions/assassins). Follow with a Noxious Blast to Ap/Ad carries (such as Annie, Malzahar, Ashe, or caitlyn]]). These are your biggest threat, and therefore your first target. At this point, you need to pick your target and do not stop firing your poison and fangs. At some point in the game, you will become heavily focused during team fights (as you are a carry). It is for when you are focused that you want to save your Petrifying Gaze/ Flash if you can. When the other team targets you, it makes it difficult for them to dodge your Petrifying Gaze; the damage is nice and the stun is incredibly useful: it is one of the longest lasting and highest damaging stuns in the game. Landing your ult will consistently turn the tide of any team fight, and if your team focuses correctly, can quickly score an ace with little or no losses.

Targeting Order:

#1-Burst Mages

• ex. Brand, LeBlanc, Annie
  #2-Assassins
• ex. Evelynn, Shaco, Akali
  #3-Ad Carries
• ex. Twitch, Ashe, Tristana
  #4-Support
• ex. Soraka, Sona, Karma
  #5-Fighters
• ex. Udyr, Renekton, Warwick
  #6-Tanks
• ex. Rammus, Amumu, Shen

Side Notes:

Don't be afraid to save Flash for escaping.
There are a few specific champions to be careful for:
Most assassins can be dangerous, but the 2 most dangerous are Katarina (her ult makes your ult not able to stun her, due to constant spinning) and LeBlanc (she is the largest bane of squishy casters, and may warrant getting a Banshee's Veil)
Ganks- you are squishy, and have no real escape moves, so you are very vulnerable to ganks all game.

Grabbing Blue buff can make for a nice combo with your early to late game. Cdr + mana regen make it even easier to spam. Red buff is 100% un-necessary
Dragon can be soloed easily once you have decent Ap (about 150) and Hextech Revolver, but back-up will always be nice.

Cassiopeia is my favorite champion to play, and is definitely the most challenging; both on the other hand, she is also the most rewarding to play. Cassiopeia is one of the least played champions, and is usually frowned upon as a feeder, but I can tell you from personal experience that she is actually one of the most fearsome of champions.

I hope you enjoyed the guide, and I hope you have a great experience playing as Cassiopeia.
If you ever not sure if you want to play as the scariest and most poisonous serpent, just remember. . .

Eventually, they all come crawling back