If you are the only living kin to a person who has died, what rights do you have over other family members?

You say that you are the only "living kin" but also write that there are other people in the family, so it is unclear what your relationship to the deceased is (are you a child? grandchild? sibling? neice/nephew? etc.) and the relationship of the other family members to whom you refer (was the deceased married? did he have half-siblings, cousins, etc.). Without such information, a definitive answer cannot be provided, so you should repost you question with additional detail.
That said, if you believe you are the closest or one of the closet blood or adopted relatives to this person, file in probate court in the county in which he lived for what are commonly called "letters testamentary," to be appointed the personal represenatative for the estate. You'll need to provide notice to the rest of the "family" that you are doing this--that provides them the chance to challenge your request, if they believe they have grounds (e.g. a will naming someone executor). If no one objects, you should be appointed and have the authority to manage the estate and distribute its assets according to your state's rules for "intestate succession" (who gets what when there is no will). If someone does object, they will need to come forward with the basis for their objection, such as a will.
A probate attorney can help you do this; if you want to proceed "pro se" (as your own attorney), you should be able to find basic instructions and possibly sample forms from the court, either online or at their clerk's office.