How exactly is Marty Byrde laundering money?

Note that Marty's laundry problem is different from the common laundry problem. What I call "common laundry problem" (the one you seem to refer to) is when you have a big amount of dirty cash and you want it to make it clean in your own bank account.

Now here, Marty's problem is different. He has clean cash of his own, and his mission, to prove his proficiency to Del, is to legally transfer the 8 million to Del's accounts.

I believe the Cartel has various legitimate business activities that include construction materials (we learn later that they do tiles) and play the role of a buffer to launder money. So they need to buy stuff to these companies to clean the money.

WENDY: I've been trying to tell you since yesterday about that half-built house on Sunrise Beach.
MARTY: What about it?
WENDY: Well, I used the money from the Chicago house and I bought it. Now, I'm no expert, but I'm pretty sure you can inflate construction costs and launder money through it. And quite frankly, I don't give a shit if you like it or not. 'Cause I feel pretty good about it. It's a good idea, and I did it for our family.
_{Tonight we improvise (S01E04)}

That's the point of the church construction idea.

WENDY: So what exactly is their [Snell's] turf?
MARTY: Well, fucking everything, right? Probably why every single business rejected me the second we got in here. Now I got that dead body. No one's gonna wanna work with me. We're screwed. We got $4 million left to launder. How are we gonna find enough businesses to clean that much without getting into a war with these people?
WENDY: We have to build something of our own.
MARTY: Flipping houses isn't gonna do it, Wendy. Could flip 50 houses, it's not gonna make a dent.
WENDY: Then it has to be bigger.
MARTY: Like what?
WENDY: I don't know. But you can move money through construction costs.
MARTY: Yeah. It'd have to be massive.
WENDY: Yeah! Well, obviously. Lots of people, lots of cash.
MARTY: And no IRS.
WENDY: So what is that?
MARTY: You tell me. WENDY: There's gotta be some business the IRS is afraid to touch.
_{Ruling Days (S01E05)}

The church builders will buy the needed materials to this buffer company, but the buffer company will write way more materials that were actually bought. For example, if the church builders bought $10000 of materials, the buffer company will write 100000$, and Marty will add $90000 dirty money so thats $90000 cleaned in the process.

He used the same trick to renovate the Blue Cat, when Rachel noticed that they wrote way more materials that actually delivered.

Spoiler if you didn't see S01E09:

When the church was burned down and he wrongly blamed Snell, he said to him that instead of burning, he could have contracted a destruction company (his buffer company) that would have overcharged the destruction and they missed an opportunity to clean more money.

1. I just earned $100,000 selling drugs.
2. I "buy" 25 air-conditioners for $100,000 from Company A. The thing is, Company A is just a shell company I own.
3. Company A now has $100,000 in clean and legitimately earned money. (And I can now use this money.)

(The above illustrates the broad idea. There will of course be many other steps that one may take to reduce the probability of audits/detection.)

The Coca-cola example does not work with large amounts of money. You can only then launder up to a 1:1 ratio of illegal client money for your legitimate customer money. That would require a lot of legitimate customers. Larger amounts of money require receipts, taxiing, and cost justification. Why would a legitimate customer pay twice the going rate for a product?

If you were all three parties (buyer, seller, and wholesale supplier), you could make it work. It would be very risky. The chances of looking suspicious and getting caught would go up exponentially to the amount of cash being transacted.

Another way would be to sell the item twice as a sort of straw purchase. If the seller and wholesaler are the same party, the seller could buy the product from the wholesaler at an inflated price. The buyer would then buy the item for a reasonable price. The seller would then be losing money. But, who cares. The only way to cover the difference in price would be to inject the illegal money of the client into the cash flow of the seller, mixing it with the legal money of the buyer. The money coming to the wholesaler would be clean although tainted. At least, it would be receiptable. Only the seller part of the operation would be at risk. Only if they hide the loss by receipting the buyer at a higher price.

Another take on the above scenario would be for the seller and the buyer to be the same party. Again, the product would be overpriced. But, the product would have to be something that could be very price flexible. Or, something that would require a contracted price due to uniqueness or difficulty in obtaining. This could be anything from art to industrial equipment. Just as long as there is clean cash from somewhere to mix with the dirty cash.

The least risky scenario would be to engage in a cash business with few to no receipts and no hard assets being transacted. Vice businesses like strip clubs and casinos are perfect for this. Dirty cash from the client can be injected into the business labeled as clean cash from the customers. If it is a business that typically sees a lot of large legitimate cash transactions. It would be harder to detect and track the mixing of clean and dirty money. This especially works when paying out more to the customers (in this case, gambling winners) more than you are taking in revenue from their business.

During the run of the show, Marty and family engage in a little of each of these schemes. An example of this is Marty buying all of the struggling businesses with cartel money. Then, all of the revenue from the businesses were transferred to the cartel at a loss. The loss was the cost of doing business to convert the illegal money into seemingly legitimate money.

Another way of doing this is to use a legitimized business like a bank to give the tainted money the air of legitimacy. For example, if Marty were to get a bank loan for a business, he could then pay back the mortgage or business loan with cartel money. It would not matter if the business made a profit or not.

Typically, the US government puts the onus on the seller and/or the lender to know and vet their clients/buyers. You are supposed to vet any cash or cash equivalent coming into or being put into circulation through a financial institution licensed by the federal government. Since vetting every small transaction would be too tedious, time consuming, and cost prohibitive, the federal government puts a cap of $10,000 on transactions needing scrutiny. The federal government also stipulates that transactions, spread out over time that have the appearance of trying to circumvent the cap, are also to be scrutinized.

Illegal operations buy property and businesses using cash and loans. They get either unscrupulous middlemen or lenders who themselves have clean reputations to handle the transactions. These “straw purchases” are still illegal. But, now the cash has the air of legitimacy.