How do I charge a client a % of profits?

If you're in a situation where profit-sharing is on the table, you could instead ask for a revenue-share.

There are fewer ways to game revenue share via accounting tricks or other shenanigans, whereas there's plenty of ways for insiders to extract cash from the business without it being classified as an accounting profit.

For example, suppose surplus cash flows generated from sales are funneled into large executive salaries and classified as operating expenses. The increased expense of executive salaries reduces the business' profit, which would lower profit-sharing liabilities but wouldn't change anything from a revenue sharing perspective.

Other dodgier ways could be related-party deals where the business enters into an agreement to buy something from an entity controlled by one of the executives -- leasing the company office owned by an executive for above market rates or so on. Cash is extracted from the business before being classified as a profit. But again, from a revenue share perspective, this doesn't change the situation.

Depends on the industry and client doesn't it. I've heard of "Hollywood accounting" where movies were intentionally made unprofitable so that percentage-of-profit based payouts are low. You'd want to adjust how you do it based on the risk that expenses could be padded to make profitability look worse.

Even if the client is completely legit, the incentive is there. It becomes like spending pre-tax money, so why not incur some more expenses.

Audited income statement

Annual P&L report should suffice.

Another idea in addition to these ideas is charge based on number of licenses (e.g. linked to usage, number of users, or so on). This is easier to track and you can track it on your end.

you ask for audit rights in your agreement