5 year lease is over

[Lex Phumirat]

Need a little help with understanding Evictions and leases.   

I'm closing on a duplex in Redlands Ca.  There is currently a tenant in one of the duplexes. He's been there for 5 years, but his lease is up in about 2 months. 

I want a brand new tenant in there because the current rent is below market. 

My question is, 

when his lease is up, does he just leave? Does he have the right to stay on a month to month?

Do I have to give him a notice first that I'm not going to renew his lease once his current one is over?

I have a 60-day notice to Vacate form, but not sure when to give it to him or when will that take effect.  Since he only has 2 months left do I give it to him right away or do I give that to him once his lease is over meaning he'll have 2 more months in the property?

thanks in advance.