Cassiopeia is a great harasser in all stages of the game. All three of her basic abilities, especially Noxious Blast, can be used constantly in lane to zone, capable of dealing huge amounts of damage over time at a safe distance.
Noxious Blast has a slight delay; lead your enemy when targeting with Noxious Blast to ensure the hit, or use it on them when they will have to stand still to last-hit.
Using Noxious Blast in the brush does not reveal Cassiopeia, so when in the side lanes, use the brush to your advantage to harass with little to no retaliation from enemy minions.
When chasing/fleeing, be sure to use your Noxious Blast on an enemy champion to speed yourself up, and Miasma to slow them down.
Using Miasma in the middle of the lane when enemy minions are coming will slow them and deal some damage giving advantage to your minions, this tactic is best used in pushes as intentionally pushing your lane during the laning phase opens yourself up to ganks.
Since Miasma grants vision over the area affected, it can be used to check brush.
Noxious Blast can also be used to check brush, as it gives you a movement speed bonus (and a buff icon) if an enemy is hit. If you cast it in a brush, and notice the buff icon, you know that an enemy is there.
Twin Fang becomes extremely powerful if the target is poisoned, allowing you to deal out large amounts of damage with its low conditional cooldown.
It is very highly recommended to smart castTwin Fang, as it will make it much easier to land the maximum amount of Twin Fang allowed by the duration of your poison effects.
If the situation is favorable, try to cast Twin Fang while directly on top of your target, as Twin Fang's reduced cooldown begins when the poisoned target is hit and not on spellcast.
Due to Twin Fang having the potential to be constantly refreshed, don't be afraid to target durable melee champions if necessary. Unlike many other mages that rely on burst damage, Cassiopeia and her Twin Fang can realistically cut through tough enemies with time, accuracy, and enough mana.
Try to use Petrifying Gaze offensively, and usually against more than one target. That said, it can also be used when being ganked to escape enemies.
It is, however, safest to use it when a teamfight has happened. The enemy will be less likely to manage to turn around with their attention split amongst five champions - and more likely to all be facing you since that is largely required to fight your team.
Petrifying Gaze also allows Cassiopeia to be a very strong anti-ganker. Simply casting it when your lane opponent and the enemy ganker are in range, then bursting down the weaker one with your combo can turn the 2v1 into a 1v1. This tactic is only effective when your Petrifying Gaze stuns both targets.
Void Staff works fine on Cassiopeia if enough enemies have a high amount of magic resistance, as the potential low-cooldown and damage on Twin Fang lets Cassiopeia be less stringent in choosing targets compared to other mages.
Getting a Tear of the Goddess and then upgrading it into an Archangel's Staff will greatly increase Cassiopeia's ability power, as well as solving any mana problems that you may have. It is also quite east to gain charges with the low cooldowns on her spells, as well as constant refreshing of her passive.
Avoid building cooldown reduction. Because Noxious Blast's damage tick is constant and can only be reapplied, and using Twin Fang on a poisoned enemy reduces the cooldown to 0.5 seconds (which cannot be further reduced by CDR), Cassiopeia does not gain the significant benefit from CDR as other mages do.
When dueling her she will put Miasma right on you, despite the constant damage from it the time needed to move out will let her spam Twin Fang to death. It's better to stay in the venom and outburst her.
Cassiopeia has excellent range on Noxious Blast and as its main harassing tool, outranging her is highly beneficial, mages like Lux and Brand have a clear advantage.
These mages can also outburst her if done correctly.
Cassiopeia is a very powerful harasser. Noxious Blast allows her to give her lane opponent a hard time, but also be wary for Miasma, as it can set you up for ganks, make landing Noxious Blast or especially Twin Fang extremely easy, or zone you from minions very effectively. Buying Boots of Speed is a good and cheap way to limit her harassment.
Cassiopeia is extremely proficient at 1v1 fights, having the most sustained damage of all mages (maybe only matched by Vladimir). Therefore it is not recommended to engage her alone. Since Cassiopeia has very high ability power scalings, she remains a constant threat to AD carries late game, since if she lands her ultimate, she can easily kill an AD carry without resistances within the 2 second stun Petrifying Gaze provides her.
Champions who can outmatch her sustained damage but also survive the 2 second stun like Vladimir, Kayle, Elise, or Swain can potentially duel and kill her.
Due to Noxious Blast and Miasma, Cassiopeia is very powerful at kiting. If she gets Rylai's Crystal Scepter she will be almost impossible to reach if played correctly; do not engage her with a champion without strong gap closers or excellent ranges while also being able to withstand her damage in a 1v1 fight.