"run deep" is usually used for a river, it means the water's flowing is very deep (meaning, the river is deep). it means there is a deep and long connection. The saying is "family runs deep", but he is making a change to that, saying "family only runs so deep", meaning, there is a limit to how deeply family members are related.

If I remember it correctly, Zach is Damon's and Stefan's nephew. Usually people trust their family members, but Zach doesn't seem to trust his vampire relatives and took the vervain as a protection against them. "Family only runs runs so deep" means even though they are related, Zach doesn't trust Damon and thinks he needs protection.

that was great, I've never heard of it. ;)