"I don't want to use any chemicals"... with 1 acre to be cleared for the first time in 30 years. Either you dont understand the task involved or you have all the time in the world to do it.
1 ... Get a fork and a spade :- use the fork to loosen the ground before you dig each sod out before breaking it up and removing the weeds etc.
This method should take many months or possibly years if only done at the weekends.
.... and dont forget to keep returning to the bits you did earlier, because they will have more weeds on them before the acre is cleared.
2 ... Hire a proper strimmer and a face guard. Then strim the whole lot down to the ground. Get yourself some of this ...
http://www.amazon.co.uk/ROSATE-36-PROFESSIONAL-GLYPHOSATE-WEEDKILLER/dp/B00CBXKFLS ... and find some one who is qualified to use it...it wont take more than a few hours.
Rosate 36 only works of plants with leaves, not bark. When it touches the ground it becomes inert.
Leave the plot for a week or so, then the weeds will show again.
Spray it again, and repeat the procedure once more.
More than likely nearly all of the weeds will be gone.
(Some weeds require specific chemicals to kill them)...
The ground is then ready to plant on.... as Sunny Dave says spuds are favourite for clearing the ground.
I have 2 allotments that are 60 sq. yds each. I find that quite an area to look after .... There are 4840 square yards in an acre that equals 60 allotments.
My first thoughts are clear it ... caravan club site, horse stables, and paddocks, chickens, free range eggs. the possibilities are endless.
On second thoughts... chickens would clear it for you. just provide shelter and let them get on with it. They will clear everything that grows.
Good luck ..!