Hello.

I wonder if it's possible to say: "He made an attempt of becoming a politician" or is it only "an attempt to become"?
Do you think it's possible to use "first" in the context: "they are making first progress in building a DNA model", and if yes, does it need "the" (the first progress)or, maybe, "some first progress"?
Thank you for all your help.

"An attempt to become a politician" is better.

"The first progress" is correct.