When you say "I like to do something", this means that you are thinking of acting on what you like to do.
Whereas, when you say "I like doing something", this means that you are at the very moment acting on what you like doing or have already done what you like doing.
For example:
I like to be a singer. - You aspire to become a singer. You are not yet a singer but you want to be one in the near future. You are looking forward to being a singer.
I like being a singer. - You are already a singer. You are living the life of a singer. You are looking back on your life and realized that you actually like being a singer.