Try to 1v1
Do NOT tower dive
Your r can be a tide turner, use it wisely
You most likely will not be able to kill tanks, stay away from them
stay out of sticky situations for escape is not likely

Hello this is Pyroclasm123. My mission is to take the under rated champions and master them. Last week I played against a couple Cassiopeias and they were just plain bad. So I decided to buy her and discovered I do quite well. She is great at hit and run because her q is delayed. this means that you can harass even while they are tower hugging. Note, I do not recommend Cassiopeia to players with little experience for, based on the number of bad cassiopeias I have seen, is hard to play for most. Note, this is my first build, please bear with me.

I use magic penetration marks, Manna regeneration seals, Cool down reduction glyphs, and flat health quintessences.
The magic penetration helps to do more damage output
The cool down reduction helps you use her abilities more often and hence increases the damage output some more
The manna regeneration lets you use her spells for a longer duration which allows you to do more damage for longer
The health makes her more durable which lets her stay alive longer which lets her use more spells which lets her do more damage.
See how this works?

I take a basic 9-0-21 caster masteries

A big problem with Cassiopeia is that she is very squishy. To counter this I get the Rylai's Crystal Scepter and Zhonya's hourglass, the health and armor help a bit. She is also very manna hungry so that's why I get the Archangel's so soon. you may want to get mercury treads rather than Sorcerer's shoes if the enemy has allot of magic damage but that is pretty much it.

I get my q first for early game harassment (see how to fight)
I then get my w, and then my e
the order to level up your skills are:
r if available
q if r not available
I then like to balance out w and e
This maximizes your combo damage

BASIC COMBO:
q,w,e,r,e,q,e,w,e,q,w,e and so on (note: you may not be able to keep this up for long unless you have blue buff although archangels staff lengthens the duration quite a bit)
MINIONS:
Use your q then w on one group (Close combat or Ranged) and they are screwed (mid-late game, in beginning you need to throw in a few auto attacks.)
Do NOT use your e too much on minions (will eat manna)
EARLY GAME:
Harass with q and w
try to stay a little back (you are squishy)
try to last hit (duh)
MID GAME:
Use your r to set up ganks but wait until they are looking at you
look to get easy kills
possibly ask for help getting blue buff
your r gives your team a few seconds of free damage, be sure to use them
don't be afraid to go from lane to lane to get kills and finish of stragglers
LATE GAME:
After a fight initiates use r for maximum team damage output
spam everything, as soon as the cool-down expires use it, her passive makes consecutive casting cheaper.
Try to have blue buff at all times because spamming can be expensive even with her passive
Use your e only on poisoned targets so you can spam it

I get Flash and ghost simply for escape mechanisms. You should choose summoner spells based on your play style.
Cleanse is also a good choice for escaping
You do not really need exhaust because her w slows and her r is an AOE stun
You do not need ignite because of all the poison
You can go ahead and use fortify, clairvoyance, or rally but I don't see why you should with Cassiopeia
I'm not even going to talk about revive
Maybe heal...
Clarity is an option
Smite only if your jungling for some weird reason...
Teleport might be a good choice because she is rather slow

In the beginning of the game you are very manna hungry so farming is rather difficult but mid to late game, or after you get your tear of the goddess/ archangel's staff, all you do is spam your q and w. This does lots of damage a second thanks to poison and lets you practically stand there while the minions die. (you still auto attack of course)