Hello Roman,
In general, using " in' " at the end of words normally ending in "ing" is used to indicate casual and informal speech in the USA principally, and would only be used in writing to quote what somebody is saying. You would never use it in official documents.
I have two further comments - one is that in your example, "gonna'" (with an apostrophe at the end) is wrong. The word is "gonna". "I'm gonna do that".
The second comment is that the apostrophe serves two purposes.
1. The " 's " indicates "belonging" or the genitive. For example: the boy's arm hurts, or the government's policies encourages tax evasion.
2. The apostrophe indicates missing letters. For example:
I'm goin' for I'm going. The "g" was missing.
The term I'm for I am. The "a" is missing.
Even "three o'clock" is an abbreviation for "three of the clock".