Is it possible for a non-Chinese private individual investor to bet on the Chinese Yuan Renminbi (CNY) going up against the U.S. dollar ?
It ought to be possible to buy a foreign exchange future (aka forex future / FX future). Businesses use these futures to make sure their exchange rate is predictable: if they put a bunch of money into manufacturing things that'll be ready a year later, it helps to know that the currency exchange rate shifts won't wipe out all their profits. If you're willing to take on some of that risk, and if things go your way, you can make money. They are essentially contracts between two private parties to pay each other a certain amount of money based on the movement of the currencies, so the Chinese government doesn't actually need to be involved and no renminbi need to change hands, you can just trade the contracts.
Note that the exchange rate is currently fixed by the Chinese government, so you're going to be subject to enhanced levels of political risk, and they may not be as widely available or readily tradable as other foreign exchange futures, so check with a broker before opening your account. I couldn't find them on my personal Etrade account, but a quick Google search reveals CME Group offering some. There are probably others.
Foreign exchange futures are an advanced investing tool and carry risk. Be sure you understand the risk, in particular how much money you can end up on the hook for if things don't go your way. Also remember, futures expire: you're not just betting on the rate changing, but you're betting on it changing within a certain amount of time.