So, are you asking just when you should submit the primary, or when you should start working on it?
You should start working on it now if you have the time. I can't tell you how many times I wrote and re-wrote my personal statement, but I know it was at least like 10. And I feel it was really a difference maker in getting me a few great interviews. Also if you have a decent premed adviser at your school make sure to schedule time with them now as well, cause this is right about when he/she will start to get the busiest and you can't just show up the day before you're submitting and hope they can give you much real input. Plus I feel like they're more receptive to students they know and have met with a few times and willing to help out those students more. Like I was in South America for all of May and finished my primary stuff while I was there and swapped emails with my adviser at least once a week to get her feedback. These people are just like admissions officers in that they read 100s of these apps every year, so they can give you a ton of help if you are really serious and stay on top of things.
In terms of submitting it, I'm completely guessing here, but I imagine through July is when AMCAS starts getting the highest volume of submissions and starts backing up a little so it may take longer to get verified. I know people that submitted in late July and it took almost 6 weeks to get verified, so keep that in mind. I'd say that to get it verified and ready to go by mid-July you'd probably want to submit no later than June 20-25th.
For this year, the earliest date was June 5th (I'm pretty sure on these dates though give or take 1/2 days...it is still early on Monday) to submit, though AMCAS did not send any of the applications to schools until June 20th, so as long as you were verified by June 20th you were among the earliest possible applications. I submitted like June 10th and was verified the 16th.
I know from personal experience the importance of applying early. Last year I was submitting secondaries at the deadlines and I had 1 interview at a school I really did not even want to go to. Well you can look at my MDapps for this year, but obviously a huge difference, and my GPA actually lowered like .03 and I did not retake the MCAT. All that changed was a few activities and I obviously re-wrote everything. I just feel like the perfect example of how much of a difference applying early makes. So good luck with everything