How can I evict a tenant whom has paid no deposit, no rent, and has no lease out of an apartment in Arizona?

The same way you evict a tenant who has paid. Arizona Landlord Tenant Law - Evictions (http://www.keytlaw.com/leasinglaw/azevictions.htm) Even lacking those documents you must have allowed the "tenant" to establish himself as a resident so follow residential eviction regs for Arizona.