Closing on a home usually takes around 30-45 days from the time your offer is accepted to when you get keys to your new home. If you want to have keys to your new home a week or two before your lease is up, then you'd need to have an offer accepted no later than early May.

If you are seriously thinking about buying a home, I'd go get pre-approved, talk to some agents, and start looking at houses within the next few weeks.