First you start of with the normal Boots and 3 pots (you can also go 2 health pots and 1 mana pot if you are bad at keeping your mana) Then you build the Sorc boots so you first of all run faster (its for the sake of your ult so you can walk right beside them). This makes killing a lot easier. then you get your cata so you dont get bursted down by the enemy. Then you should build Chalice to help out with the mana problem when you dont have blue. After that get your Rod so you got some more health, mana and ap. It also helps when you level up and you are low on health and/or mana. Then you upgrade Athene's (you might have to build the other parts I would start with Fiendish codex if you cant get a direct Athene's). This Helps you even more with the mana and you get some cdr so you can spam your spells even more and who doesnt love a Swain who can spam his spells. Then the Giantbelt to make you more tanky. Swain is a tanky mid carry. So you should get Giantbelt. Then you should get the Blasting Wand and the book to give more ap so you can finish your Rylai's. This is all to make you more tanky. Then you build the Blasting ward again for more damage and then then the negativ for magic resistant. The you build the Abbysal so you are more tanky vs ap and you and if you have another ap user on your team damage more. Then you need the glacial to make you even more tanky and then build it into the frozen heart. Why because it gives cdr mana and armor. It also slows the enemy.

Early game you want to stay back and just afk farm (or if you anti the enemy then you should just harasse them with first e then q) until level 6. At level 6 you will be pretty much unkillable becuase your ult gives yo so much health. So first e the enemy so that you give them more damage then q them to slow them and then w to lock them. You should have your ult up and running the whole time until they flash away or they are dead.

Mid game all you got to do is kill the highest dpser (mostly it will be ether the ad carry or the ap carry). First if you get with in range then use e to do more damage with other spells. Then use your q to slow and then w to lock. and then just run right next to them with ult. Swain is easy to play when you first begin to play him a lot. You just need to know that all you gotta do late game is the use your spells in this order r e q w thats it
spam that and you will surly win