Since we can't custom order willingness to help as a character trait, you'll need to find a way to convince your family that they want to do their part and that it's essential to a happy family.
Take an accounting
Take an accounting of what your family members do (without asking) to help the family. Get a piece of paper and write down each name and next to each name write down what each does, including yourself. You may be surprised to find that your spouse is, in addition to working out of town five days a week, mowing the lawns, keeping the cars maintained, climbing ladders, doing the bbq'ing two nights a week, hauling laundry to the washer and your son and daughter volunteer to help and do their chores with minimal reminding because you've raised them that way.
You need to be honest in your accounting. It doesn't make it less of a contribution just because because mowing the back yard is something he does without asking. It needs to get done and he enjoys it or takes it on without grief. Don't spoil that.
What Is Left to Be Done
Now that you've taken an accounting of what you are doing and what your family members are doing, decide fairly what isn't getting done at all, the way you want, or divided fairly. Make up a list of these items
What Cleaning Can a Child Do?
Just because your duties may include "everything" that doesn't mean you should do everything, especially for your children. In fact, doing everything for your children may be faster and less grating on the nerves in the short run, but you and everyone they'll be in contact with in the future will be suffering for it in the long run. If your child doesn't get chances to learn skills, how will they ever learn them?
Don't underestimate your child's abilities; they are often able (and willing!) to perform many helpful skills. If you start teaching them at a young age, you'll be surprised at how enthusiastic they are and how well they get the job done when they're older. Often times if you work with your child in the same room, it will stay someone enjoyable and will get done much quicker. Don't forget to turn the music on and let them wiggle while they work!
Age appropriate skills:
Ages 1-3--Room pickup. Provide big and small baskets for everything, one for stuffed animals, one for legos and blocks, one for junk to sort through twice a year, one for dirty clothes, etc. No time minimum or maximum.
Ages 4-7--The above plus: House pickup. Bed making. Dusting of low shelves. Sweeping. Vacuuming. Laundry pickup. Folded laundry put away in their own drawers. Kitchen table clearing and wiping. Quick bathroom floor clean (give them an already wet towel and have them wipe the floor with it once a week; it keeps the scrub job you do later from being too gross). Chores can be put in writing at this age, letting them pick between chores is better. Any cleaning beyond their toys and their personal messes is 15 minutes a day maximum. Set the timer (better yet, let them, they love it).
Ages 8-10--The above plus: Dusting of high shelves. Laundry, all stages. Emptying the dishwasher. Helping with dishes. Cleaning glass windows and surfaces. Begin teaching clutter control and organization, starting with separating school work and toys into their own areas. Any cleaning beyond their toys and their personal messes is 20 minutes a day maximum. Set the timer.
Ages 11-18--The above plus: Start teaching new chores they will use on their own. Show them how to make cleaning supplies on their own, what the basic supplies are, explain why some are dangerous, etc. Have them help you find bargains shopping, find coupons in the paper. If they get good enough at home, tell them you will allow them to make money cleaning others' homes if they wish. Responsible for their own bathroom cleaning, twice a month at least. Dishes and kitchen tidying 1-3 times a week. Ironing their own clothes, if necessary. They should also start learning about home maintenance, such as bill paying, filter changing, garden and lawn care, utility shutoffs and emergency readiness. Any cleaning beyond their personal messes is 30 minutes a day maximum.
Try to Make It Fun
Children love games. Try making cleaning into an afterschool game. Sure it takes some time on your part, but it will be fun, the job will get done, and there won't be any yelling for a short while.
One of my kids' favorites is "Attack the Mess." I will hide a favorite snack or fun ticket (such as YOU WIN A FREE HOUR PAST YOUR BEDTIME!) where the mess is and give them instructions like a treasure hunt to find it, with instructions on what to pick up on their way. Make it worthwhile for them!
Children love gadgets. Ever since I got my cordless small vacuum cleaner and cordless carpet cleaner, my kids volunteer to use them. Of course, you have to control their use of it because once they realize it's only another cleaning tool and not a fun toy, that'll be the end of that.
Children also love schedules. If they know they can play, play, play all day without cleaning up but that they must cleanup from 4:00-5:00 pm every day, they can understand that much better than sporadic requests to clean up games they just can't bear to pick up yet.
Children love earning money. I don't give my kids an allowance but I do pay them for doing certain chores. These are chores of mine which I don't want to do, like weeding or mopping the bathroom floors. In addition to getting the yuck chores done and having free time for myself, my kids are learning the value of money and that you don't get money for nothing (an allowance).
One other quick tip: make one large or several small shelves in one spot--call it the family game shelf--and in it include a small basket. All games and puzzles must be returned to this one spot, with any spare parts or pieces put into the basket for sorting later. This is so much easier than having games spread out and stored in every room, plus I've found my kids like knowing where everything is and clean up quicker and with no fuss.
Till Dust Do Us Part?
After selecting him carefully, my next secret to having a more helpful spouse is compromise. Until you're willing to sacrifice your size 10 silk shirt and see it turned into a perky size 2, you won't be able to show him that helping is worth it. That doesn't mean you must sacrifice all your clothes, but you must give him a chance. I've since started wearing less destructable clothing and showed him where to pile any clothes he's not sure about.
Laundry at our house is washed separately when the whites no longer look white and then I do a big bleach load. Otherwise, everything goes in the same load. Being picky about the laundry was an area I was willing to give in on because if I didn't gripe about how he did laundry, he would haul clothes to the washer, start loads, and pile the clean clothes on the couch. You all have a couch like that, or maybe it's a room corner, which everyone avoids because no one likes to fold laundry!
Where can you compromise? Would you be willing to live with the toilet seat up if, once a week, he sprays toilet cleaner in the toilet and used a long scrubber to swoosh it around? "Honey, how 'bout you use this toilet cleaner now and then and I'll stop grumbling about the toilet seat up?" Corner him in the bathroom and show him how easy it is. Humor to get these things done is very effective.
Bargaining is also a useful method. Try trading or bargaining chores for one another or for favors. "If you do [chore] for me, I'll do [chore] for you" or "Hey honey, hold this broom and sweep for me! I'll make it worth your while."
A spinoff of this is something I invented years ago...betting! Do you make bets or deals with your family or friends? Betting with money always seems stupid and wasteful, so I devised a way to not only "bet without money" but find out what is really important to your spouse, kids, or friends. Here's how this goes at my house:
Me: "Honey, let's play cribbage."
Spouse: "OK, what's the stake?"
Me: "20 minutes a game."
Or, bet who will win the World Series or what time the next phone solicitor will call. For my kids the bet is something like "I bet you I can do 20 situps in less than a minute, if I win, you have to pull weeds for 10 minutes, if I lose I have to make Rice Crispy treats." Use your imagination whenever the urge hits you or whenever the floor needs mopping and you just can't handle doing it one more time.
What does betting minutes mean? Well, if you win, he gives you 20 minutes of whatever you want. Whatever I want has been "pulling out ivy in the parkway, folding laundry, backrubs, foot massages, etc".
I bet you think you know what your spouse will want with his won minutes. I've been very surprised by mine. It's amazing how desires are clarified and fine-tuned when you have 90 minutes of "whatever" stored up. Be careful, if you are a known game loser, you'll never get that laundry folded.