In The Groundwork of the Metaphysics of Morals, Kant's categorical imperative says: “I ought never to act except in such a way that I could also will that my maxim should become a universal law” (4:402). In this context, how does Kant show that making false promises is immoral?

Throughout his voluminous writings on moral philosophy, Kant emphasizes the point that maxims of morality must be capable of being applied universally to everyone, in all times and all places. There must be no exception to the moral law; this is what gives it its compelling force, and to which we are duty-bound to assent.
Kant recognizes, of course, that there are situations where, for whatever reason, we cannot act according to the dictates of the moral law. But he argues that, in such cases, we are not acting morally, for the simple reason that we can't universalize our actions so that everyone would behave the same way in a similar situation.
Kant uses the example of lying to illustrate his point. Whenever we lie, we are effectively willing that lying should become a universal law. Kant wants us to think through the very serious consequences of this. For if lying does indeed become a universal law, then imagine what it would be like. Promises would mean absolutely nothing, as there would be no standard of truth against which they could be measured. Relationships of all kinds, be they personal or business relationships, would be virtually impossible to maintain. People would feel free not just to lie, but to cheat, steal, and obtain what they wanted through making false promises. The very fabric of society would become dangerously unraveled.
Although Kant thinks that lying is intrinsically wrong, that is to say wrong in itself, he also recognizes the damaging consequences that would arise in the event of lying attaining the status of a universal law.