By far the most common way (the only way?) to enter Abkhazia from Russia is through Sochi. For a while foreigners (except Russian citizens and members of UN) were not able to enter Abkhazia. I don’t know when it has changed, but it is not a problem now — just make sure you have multiple entry Russian visa. The only way out of Abkhazia is back to Russia, as transit to Georgia is not permitted by Abkhazian government. Actually, I’ve heard of charter boats between Abkhazia and Turkey, but I don’t know what’s the deal there.

This good information shared Sergei Bagapsh, the President of Abkhazia during an interview to the Russia’s biggest online news source Lenta.ru

Sergei Bagapsh: Citizens of third countries can enter Abkhazia with out any problems. If a tourist has multiple entry Russian visa, he can come to Abkhazia with it as well. Many foreigners visit Abkhazia these days.

Link on topic: Official website of the President of Abkhazia.