All campers "using the bushes" as their toilet need special camper's toilet paper and a trowel (a tiny shovel like gardeners use). And assuming that you are female, a "Go P" type device is also very helpful.
For "the small things" you just go behind a tree or bush, empty your bladder, clean yourself with the special toilet paper. You can dispose of the used toilet paper by burying it -- that's why you have that trowel -- or by adding to the collection of other trash you are packing back out of your site.
For "the big thing" you dig a "cat hole" first. That's just a hole in the ground, You "go" into the hole then clean up with the special toilet paper. Put the used paper into the hole along with the "go." Then fill the hole back up with the dirt you dug up.
If your group is camping more than 1-2 nights OR if your group is bigger than two people, then your group needs to dig a latrine for "the big thing." This can be as simple as a super-size cat hole with a good sized pile of loose dirt next to it. After each camper "goes" into the hole, that camper shovels a bit of dirt over his/her "go." When the group finally leaves that campsite, be sure to fill the hole completely with dirt before going.
The "Go Girl" is one type of "Go P" device for women. It's available from Sierra Trading Post.
Campers' Toilet Paper (also known as "mountain money") is available from most providers of camping goods.
I recommend also having sanitizing hand wipes for a good hand cleaning after "the big thing." And remember, those sanitizing HAND wipes shouldn't be used like toilet paper. You need baby wipes, designed for use on that part of your body, if you want moist clean up in that area. Be sure to dispose of wipes properly -- if you pack it in, then you pack it out too.
An empty plastic jar with a wide mouth and tight fitting lid can be a good way to contain yucky trash items.