Unfortunately, the best time to plant grass is September/fall. That gives enough time for the newly formed roots to get deep enough to withstand summer heat and drought conditions.
Spring grass will germinate and look great until the summer heat has at it. Then, in my experience, a lot of it will burn up and die. Plant some rye, see what happens and then be prepared to overseed in the fall with a mix containing fescues, bluegrass etc. A mix is good because one of ingredients will likely be more suited to your local conditions (soil, sun etc) than the others so that one will eventually take over.