JK's and pop-ups are a great combo.
On the technical side, look for either the "dry" weight or "curb" weight of the camper. This should be listed on the RVDA sticker somewhere on the vehicle. Dry weight is the lightest weight of the camper without you adding anything to it. Curb weight is the weight of the camper with things like you filling the water tank (if equipped). The JK has a tow rating of 3,500 lbs; however, if you attached a 3,500 lb trailer to your Jeep and then throw a cooler, some misc. gear, and your family into your Jeep you will not like the results.
I have an 08 JKU Rubicon with the tow package and my camper has a dry weight of 1,689 lbs. With all of my gear, myself, and a passenger or two, it tow just fine most places. Where it brings the suck is going up grades in the mountains. At 7% I have a hard time maintaining 35 mph.
For what you are describing as a requirement, I would use 2,000 lbs or less dry weight as a good rule of thumb. Luckily there are plenty of pop-ups in this range.
The first thing I would consider in the floor plan of the camper would be sleeping arrangements. Our camper has a queen bed at one end and the next size down at the other, a dining table that turns into a bed of the same (second) size and another table that turn into a small bed. With two young kids you could probably get by with just the two end beds but having a third bed would be convenient for when they get older or they bring friends along.
Next, having a heater is a big convenience. Even in warmer parts of the country it can still get cold in the middle of the night. Setting the heater at 60 really takes the edge off the air and makes for a comfortable nights sleep when it's below 40 outside. A heater will also require a source of electricity.
Most campers will have at least a 12V system. Having a power converter is nice also. This will allow you to take advantage of a 110V power source if available in the campground. I use the largest marine battery from Walmart on my camper. Mine has lasted been on my camper for 6 years now and is still going strong. Not bad for a $65 battery. If we don't run the interior lights all night long, we can get through a 4 day weekend even with the heater running at night.
Propane is another thing to consider. Most campers will have at least a single bottle system. Our camper came with a single bottle and I upgraded it to a 2 bottle system. With a propane refer, heating, cooking, and dish-washing (heating hater) we can through almost an entire week.
Many campers come with small refrigerators these days. Our camper has a 3-way fridge. This means it will run on 12V, 110V, and propane. If we are at a campground that has power hook ups I will run it on 110V. If we need the fridge and don't have 110V power available, I use propane. It actually works pretty good on propane. I never use it on 12V. It doesn't get as cold and it drains the battery way too fast.
Most campers come with a water storage tank but a lot of them do not have a water heater. Our camper is this way. We never use the water tank. We just fill a 5 gallon plastic jug at the campground. I'm considering installing a small 12V pump in the camper's system but I will also want to put a faucet on the outside of the camper. (We do almost all of our cooking and cleaning outside of the camper.) I am currently using a Zodi portable water heater. We used to heat our water on the stove in a coffee pot. If you can find a camper that has a water heater, it would be convenient but I don't consider it necessary.
Most campers do not come with air conditioners. Many have hook ups but you have to pay extra for the air conditioner itself. I know some people consider these necessary but I don't. Without a source of 110V power they are kind of pointless anyway. People who run a noisy generator all day just piss me off and I don't want to be like them so I don't think I'll have A/C for my camper anytime soon.
I second the craigslist suggestion. We got our camper for about $3K.
One of the better items I bought was a mini sway-control set up. My Jeep actually rides and controls better at 75 mph while towing than when I'm not thanks to it.
Camping stuff I use that makes my life easier (these are just links so you can see the items, you might be able to find the products cheaper elsewhere):
There is a grill box available for this stove that is great to have also.
Outdoor Camp Kitchen:
There are lots of outdoor kitchen available. This is where we do our dishes as well as food prep.
Portable Water Heater:
This is new to me and I'm still messing with it to get it dialed in.
Last but not least, my camper:
Original Set Up:
New Set Up: