If you etch the information onto plates, it should last until some external event such as a volcano or meteor happens to hit it, and that’s why you bury it under a protective layer and have more than one time capsule. You probably want to launch capsules onto other planets and even out of the solar system, where they might with luck survive more than the five billion years that the sun will, although any that we sent into deep space, we'd just be guessing where to aim. We’ve had the capacity to do that kind of thing for almost fifty years.
As for the genome, it would definitely be possible to record the letters of a human’s genetic code, although we’d need thousands of complete sequences to allow anyone to in theory create a viable breeding population, and of course there isn’t a single human genome, but one in the nucleus and then the organelles.
Communicating that information in a way that would allow an alien species to actually create more of us is an interesting puzzle, though. Imagine receiving a specification for a self-replicating industrial robot in the form of the program you would have one of those robots run to build a new robot. In order to build an authentic robot following those instructions, you would first need to create a robot that could follow the same instructions as the original robots in exactly the same code! So we might have to try to cryogenically preserve some human zygotes and hope that alien scanners can figure out how those worked. Unfortunately, we couldn’t tell the aliens how to make an artificial womb capable of growing human clones, because we don’t even know how to do that ourselves. We might be able to describe the requirements the clones would need for nine months, and hope they can figure it out. (Would we need to include samples of blood cells and tell them they need to provide lots of those, but alive? Or could they get by on some kind of artificial blood substitute? Not by 2016.)
But even those wouldn’t mean much to an alien with a different biology, though: we might get across, using chemical diagrams, which four amino acids correspond to the letters, but then aliens whose genetic code is different would be wondering why we’re preserving long meaningless strings of amino acids that don’t represent any functional protein. So we might be able to indicate that each three-letter block codes for an amino acid in a protein. But you can’t just list the proteins that make up a human, mix them up, dump them out, and get a human.
So there might not be any way to get across, “Hey, here’s how you can make more of us if you want to,” short of cryogenically freezing some human bodies (a pregnant woman?) and hoping they survive intact enough to reverse-engineer.
What we might be able to manage is to save some spores, freeze them, and shoot them into deep space so that, if they ever thaw out on an exoplanet, some living thing from Earth might survive. But would we want to? Even if we could get aliens to create a human baby someday, with no idea how to raise it or what to do with it, would that even be something we’d want? Just trying to preserve as much of our culture as possible seems like a much better idea.
