It sounds like it could be a battery issue, but on a phone this new it seems unlikely that it would fail so quickly. At this point I would be trying to check the charging port first for problems; does it say it's charging when you plug it in?
How long have you had the phone? It should have at least a one year warranty, so it may be worthwhile to check with Samsung as to whether this can be repaired under warranty before you start digging into replacement parts.