It will cost you a little up front but you can offer some money for college/high school kids/whoever in the area to clean up trash in the area and pay them by the bag or per hour or whatever you think will get you the best return. 

I get trying to clean up the city and prevent crime but taking it on landlords probably isn't the way to do it. 

If you can charge a non-refundable cleaning fee, that would help toward the clean-up if you were to end up with the unit empty and had to rehab it, or you decided t clean the carpets or whatever.

I mean we can't force the current tenant to clean up How would you approach this situation?

Yes, because you have to give a 24-hr notice, the offending tenants typically just clean up for your arrival.

I suspect you'll need the $ for clean up some day...Maybe if they are selling them you could have a cause to evict based on running a business out of the house?  

Mark Graziano , as Jason Bott said, this is why the tenant needs renters insurance and our lease strongly recommends it and strictly says landlord not responsible for damage to personal property.Your insurance company should cover cleanup and remediation.

So just a little clean up and maybe you'll get some more calls.

Focus on cleanup and getting it re-rented.

All the belongings would normally be handled by the Abandoned  Property regs for your State, but as family is already aware, I would ask them to remove all the contents within 30 days.When one party to a contract dies, so does the contract so the lease is no longer in effect and as soon as the contents are gone you can cleanup and seek another tenant.Prepaid rents you hold are 'unearned' and the deposit should be handled as any other move-out.In Calif, a landlord must notify the next occupant that there was a death in the unit -- verify your State requirements.