Are there any good recordings of Genesis being sung or chanted in the original Hebrew?

Here's a collection of the whole Hebrew Bible in a very correct* Modern Hebrew pronunciation, without any cantillation. If Ashkenazi** pronunciation is more your style, check this out, organized by weekly Synagogue reading.

*I.e., according to the rules of the Hebrew Academy.

**Edit: specifically, Lithuanian, given the pronunciation of /o/ as [e].

Second edit: noticed the first link doesn't just have the Torah.

You'll be best off searching for the Hebrew name for Genesis, Bereshit. (Sometimes spelled "Beresheet" to avoid unfortunate mispronunciations.) "Bereshit trope" or "bereshit leining" are probably good search terms. There are a lot of recordings of Torah being read, because typically Jews who volunteer to read a Torah portion in synagogue will listen to recordings as part of learning the portion.

A couple of good sites I found just messing around on Google:

http://www.bible.ort.org/books/torahd5.asp?action=displaypage&book=1&chapter=1&verse=1&portion=1

http://torahreading.dafyomireview.com/