Max payload of the new f150's seems to vary by more than 1000lbs. Exactly which f150 are you getting? Example, if your payload is ~2200lbs:
"Claimed" dry weight of 865 you're considering (with no AC, zero options)- 2011
Propane, battery, gear, food and water for 3 people - 1000lbs
Three people = 450lbs?
You'll more than likely be at least 3500 lbs loaded. I've got zero problems pushing my payload but exceeding your payload by 65% is excessive.
To get the highest payload (which you'll still exceed) in a 150 you wont' be able to get the ecoboost engine which negates the (arguably) biggest reason to get a half in the first place. I would be considering 3/4-1 ton trucks. Driving a 350 at 75% of it's max payload is a much better experience than driving a 150 that's maxed out.
Also, what type of camping do you do? If you hang outside all the time no matter the conditions and just come inside at night to sleep then sure an 865 would work for 3 people (4 in a pinch) but if you plan on spending much time inside (rain, mosquitos, whatever) it'll be VERY tight. Getting a long bed 3/4-1 ton would give you a lot more camper.
I say all this because your title is "help me do this right". Can you drive a severely overloaded f150 and cram 4 people in an 865, sure, but I wouldn't say that's "doing it right".