"The key to becoming an expert..." OR "The key to become an expert..." ?

The correct form is “The key to becoming an expert….” The reason is that key should always be followed by a noun, and becoming is a gerund, or noun form, of the verb become.

Here are some additional sentences with key followed by a noun:

• The key to improving your English is practice.
• Having a job that you enjoy is the key to happiness.
• The key to avoiding the common cold is washing your hands frequently.

The noun secret, which is a synonym for key, behaves the same way:

• The secret to becoming a good writer is to read.

However, the noun way, which is another synonym for key, should be followed by a verb rather than a gerund, even though it can be followed by other kinds of nouns:

• The best way to travel is by train. (way + verb)
• There is no easy way to say this: You have to leave. (way + verb)
• The way to her heart is to appreciate what she does. (way + noun)

So unfortunately, there doesn't seem to be a general rule that you can follow to remember whether you should use a gerund or a verb after nouns like key, secret, and way. My advice is to remember an example sentence for each one.