The claim that 0.5% of people (8% of Central Asians) are descended from Genghis Khan comes from a 2003 paper looking at the genetics of Central Asia. They found that the variation in Y-chromosomal DNA (which is passed from father to son unchanged except for mutations) is consistent with a common source about 1000 years ago. The connection was then made that Genghis Khan and his heirs likely fathered a lot of children in the 1100s and 1200s, which is a plausible explanation for a common male ancestor. The authors went a bit further, for example establishing that groups that have an oral tradition of descent from Genghis also have a high propensity of this Y-chromsomal group.

The supposed "Genghis Khan gene" is part of Y-chromosomal haplogroup C2, which is common among Asians and Native Americans (indicating a common male ancestor more than 15,000 years ago). If your genetic profiling website (23andme or whatever) says you have a different Y-chromosomal haplotype, you are not part of this male lineage (if you are a woman, you will need your father or brother to do the test). If you do have C2 (or one of its other names), then there is a cluster of sub-haplotypes that are associated with this common ancestor, sometimes called C2b and its subgroups like C2b1a3. Note that there are a few different naming conventions for these haplogroups that your service may use.

Now even if your genetic test links you to the chromosomal group that Zerjal et al. associated with that common Asian ancestor, this does not prove you are a descendant of Genghis Khan. Assigning the common male ancestor to a specific person is speculation. The way to more definitively know is to find the body of one of the royal Khans and extract its DNA, and if possible compare that to a modern claimant of royal descent (that would be very difficult after 800 years, given the propensity for false paternity over that period). This for example was recently done to figure out the Y-chromosomal haplotype of Napoleon, where a hair was taken from his crypt and used to sequence his DNA, and that was compared to one of his great grand-nephews. However, given that the Mongols allegedly buried their khans in secret unmarked graves, this is unlikely.

edit: As /u/Vespertine points out below, the whole common-medieval-Asian-ancestor thing isn't as clear cut as Zerjal made it out to be.