I'm Korean and don't have a US address or credit card, but I want to buy content from the US iTunes store.
I have a PayPal account connected to a Korean bank. Is it possible to add funds to a US iTunes account with PayPal?
Yes, you can only use PayPal with iTunes US, and in that PayPal, you'll need to have a US bank account or have your locations and any credit cards/debit's on there will have to be located in the US as iTunes checks for that as well. It all has to be in the US, and cannot have any other bank or CC company for any other country.
I don´t think using PayPal with a Korean bank is possible (haven´t tried this, of course), but if you still want to do this (even though this is probably not legal!), you should look on eBay or similar Sites for iTunes gift cards. Then you could theoretically create a new account in the US Store by trying to buy something that is free and then clicking on "create new account". Only this would enable you to create an account not associated with any credit card. Next you could activate your Gift card and buy from the US store. Note that you would have to lie about having an actual US address, which will make this probably illegal (or at least breaking the terms of service).
As an alternative solution, shop the iTunes US Store whilst in an oversea country with Jerry Cards. Receive high quality scan of the gift cards in your email upon verified payment in your email account, with Paypal or most international credit cards.
Use ebay.com and ask seller to email code. I have done this many many times.
Be warned do, in the last year or so also scammers have found a way to make money by buying cards on eBay and then telling the seller it doesn't work (when it did). This has raised awareness on eBay sellers and is one of the reasons people have stopped with just emailing you the code making it much more expensive and time costly to buy this way.
I found out that mailing the persons who just added the action telling them your intentions and stating you want to pay with PayPal and are willing to pay the full amount, works the best. Too bad in one way (no more US iTunes Cards from $100,- for $80,-) but seems to work the best. Going after a cheap one takes much more time and hassle (on selling you the code instead of the card with code by airmail).
Change your "store" in iTunes to Korea since that's where your bank is located. Your money will sit on your "account" not in that "country". Then change your store to the US store and tada, your in biz. It's completely LEGAL people trade cash like that all day long. But I do warn you to check exchange rates, the Korean currency doesn't equal the US currency value. Having an "address" just means the place your at now. ie apartment is fine, just no po boxes, I know several people that have used the local apple store as their "address". This is far from lying & it's none of Apple's biz anyway.