I currently owe my landlord 545.00 in back rent for the months of April and may. I have paid in full my June rent , my question is can he file an unlawful detainer? I live in mn.

Actually no, when you paid money in JUNE, that money would pay back rent first. So no you did not pay June, you paid for April. And of course if you are still two months behind in your rent, and you have not made a plan to bring paymennts up to date, they can file to have you evicted.

As Fr. Chuck as advised... any money received will be applied to back rent... not the current months rent. So legally you are behind for May & June's rent. Have you received any notices/court papers? It sounds like you have a very lenient landlord. I would suggest getting caught up asap and be grateful you haven't been thrown out yet. Most landlords would not let you get this much behind.