Do you say "I'll become 60 years old" or "I am becoming 60 years old?"
The simplest form would be:
I'll be 60 years old next year.
Another option, although a little more awkward* in my opinion, would be:
I'll be turning 60 years old next year.
I will turn 60 years old next year.
You could also shorten both phrases by omitting "years old", and you should be understood:
I'll be 60 next year.
*The "turning" construct sounds a bit more awkward to me, but it can also be slightly more precise, depending on context. In the general sense, either "I will be 60" or "I will turn 60" will convey the same meaning, but in certain specific contexts, they can be different.
In the strictest sense, saying "I will be 60 next year" simply says "at some point during the next year, I will be 60." This could be taken to mean that you will start the year, Jan 1, with the age of 60, and some time later, will be 61 years old.
This type of confusion is only likely in specific, technical contexts, such as retirement planning, or eligibility for an after-school program ("Participants must be 12 years old before Sept 1"), etc.
But to avoid such confusion, if it matters in your particular context, you can use the "I will turn 60 next year" form.