My method for this (similar to paddling the sloughs around SF Bay where knee deep mud is a minimum) is to put the boat stern on the last amount of dry land one can find and the bow in the water. Straddle the kayak (or sit on the stern with legs in the water) and shimmy up to the cockpit. Slide in the legs and then drop down in the seat (worked better when my knees were more flexible).
If the water deepens quickly so feet can’t reach the bottom at the cockpit, it becomes a tad more challenging. This can be done in a few ways. From easiest to hardest:
-paddler 1 enters boat, paddler 2 pushes boat. Paddler 1 goes home :), or…
-put paddle floats on both ends of a paddle. Strap paddle to boat (just behind or in front of kayak) so that you have a float on either side of the kayak (quick release straps work well for this and put the clips at the top so they can be easily removed). Shimmy to the front, enter, release paddle. Labor and gear intensive, but it works really well and even lets one wash the muck off before entering the boat.
-Face to the stern. Put boat on water, extract one leg from muck and straddle cockpit. Keeping body low, lay across rear deck. Slowly put extracted leg (and 5 lbs. of smelly bay bottom into the cockpit). if 2nd paddler is available, they can help steady the boat, btw. Extract 2nd leg from muck and slide into cockpit. Rotate body on rear deck and slide into cockpit. Put on spray skirt to contain the stench of the muck. This works, but the shortcomings are obvious and those not comfortable entering this way may have considerable difficulty executing this.
-enter water with boat - do a routine deep water self or assisted rescue with each paddler in the group (much less mud this way). There are conditions where this is the preferred method and it requires everyone to have decent immersion protection if the water is uncomfortably cold. It has the benefit of forcing everyone to practice their skills at least once a day, but if one wants to stay dry, why go out in a kayak?
You are going to get wet and (probably muddy) with each method. The last has the benefit of not filling the kayak with that awful bay bottom. Generally, this is what I do since I’m usually the last to launch. I use the first method to get everyone else out in the water. I also wear a wetsuit since it sheds mud better than other outer wear and doesn’t allow any of the muck to reach my body.