Building a miniature from scratch is very fun and as of now, I'm not spending a single penny on GW products. This guide is not so much about telling others how to do their stuff as much as it is for me to try making a step-by-step guide for the first time! There are so many skilled hobbyists around this forum that I almost feel bad about posting it but hopefully you now understand my reasons

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So, how to build your own undead chariot step-by-step:
First of, the main body. You need some stiff paper (cardboard is to thick) and some light wood. I've used a corn flakes package (tea or candy sometimes have similar packages) and an ice cream stick. I guess that balsa wood and plasticard would be better alternatives but the material I'm using can often be found in any household! I'm not trying to encourage anyone to eat ice cream but once in a while after having one just save the stick

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It shouldn't be hard to cut a chariot body of the right size so long as you have an infantry or something to make it fit. Glue the paper together to form the base and I also used some paper tape to stabilize.
Then cut (a scissor works) the stick into pieces roughly the height of the chassis. It doesn't matter if the pieces are uneven as it only reinforce the image of a rough chariot. Just look at the Orcs and Beastmen

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The pieces are cut into smaller pieces resembling "planks" that can be glued, using wood glue, to the outside if the basic chassis. It should resemble something like this:
Second, the base. Very easy, just find some suitable steeds of your own choice and either use a GW 50x100 base or a custom made as I've done.
I glued together a 25x50 base with a leftover 50x75 base onto a piece of stiff paper making it into a 50x100 base. This is because if you try to glue the bases together without something under there is a risk that the base will be unstable as it may not be parallel with the flat surface of the gaming board. When the model is finished and ready to base it is fairly easy to cover up the "crevice" made from this technique. As you can se there is a small stripe of plastic connecting the steeds. The stripe was cut from a butter package and it is as strong as regular plastic but very bendable, making it good for this type of detail. I guess a shampoo bottle could have the same bendable plastic for those not using butter in plastic package

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Thirdly, the wheels (this ones for you
@Borgnine 
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Once again I used cut out stripes of bendable plastic from household packages. Combined with some metal wire, round miniature shields, some suitable buttons from a coat or the like (they are usually super cheap at second hand store but shouldn't be to expensive at a regular store either) just make sure the buttons match the size you want for your wheels. Last, a wooden grill stick (toothpick could work but I felt it was a bit small) cut after roughly the width of the chariots main body but a little longer so the ends stick a bit outside.
This stage is where I made a mistake in making the arm to wide so it looks like the chariot is balancing on the edges of the base

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The wire go through the holes in the buttons and is twisted around the stick. It gives good stability and allows the possibility to slightly adjust the wheels up to the point when it's time to glue. Glue the plastic stripes around the edge of the buttons. When the shields are attached it will look very much like an actual wheel.
Fourth step. Use the rest of the grill stick to form pieces like in the picture. The smaller ones are attached under the main chariot body in the front with some space in between so the main arm better attaches between the steeds and the chariot.
Fifth part. Use bitz like shields or skulls to decorate the chariot. This time I also added a horn to represent the musician in the chariot unit. Any type of undead crew will suffice.
The finished chariot! These were very fun to make, crazy cheap and a good way to "recycle" junk lying around the house!