1. You are a learner. So I respectfully suggest that you follow the rule:
The adverb "not" goes before the infinitive "to do."
2. One expert says that a sentence such as "I asked him to not do it" is "weak." She says that a sentence such
as "I asked him not to do it" is "better."
3. I am sure that you have heard about the famous English writer Shakespeare. Well, some of us remember the
rule by recalling these famous words of his: "To be, or notto be: that is the question."
4. Many teachers in 2012 would NOT call "I asked him to not do it" incorrect.
(If you are taking a test, however, it would be very wise to choose "I asked him not to do it.")
5. Many good writers have "split" (divided) the infinitive. Here are two examples:
a. (from the year A.D. 1454) "Y schall ... swere to not discouere hem." ("I shall pledge myself to not inform on them.")
b. (a more modern example) "How could people be so insensitive as to not know they've got wax in their ears."
Painless Grammar by Rebecca Elliott, Ph.D. (Google books) A Grammar of the English Language (Vol. II) by Professor George Oliver Curme, page 460. The Oxford English Grammar (1996) by Professor Sidney Greenbaum, page 583.
P.S. When you have time, you may wish to google "split infinitives."