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Re: Literally dead
Well, it could be a defective battery... Is there anyone else who could lend you their S4? Try your battery in theirs, if it doesn't work, probably your battery is dead and that's the culprit (get it replaced). But if it doesn't, now it's time to try the other battery on your phone? If it comes to life, go to Step 2. If it doesn't, go to Step 3 (and probably cry).
If the battery worked in the other phone and the other battery worked in yours, it could be a faulty USB port, which would also explain why it's not charging. This is fairly easy to repair/replace and not that expensive, and my experience with past Samsung devices is, this port does tend to go kaput (S2 and Note 2, and a friend's S3, all the same issue, all the same port replaced).
If the other battery didn't work in yours and/or your battery worked in theirs, it could probably be the motherboard. This, sadly, is very expensive to replace unless you're under warranty. This and the screen are the most expensive parts of your phone and will not be easy, cheap, or fast to replace if you're out of warranty.