The most common and most likely answer to this crossword clue is the 5 plus 5 plus 4 letter word GOING GREAT GUNS
I've seen this crossword clue in The Daily Telegraph
G
O
I
N
G
G
R
E
A
T
G
U
N
S
Buddy explains!
The straight part
We know that "becoming very successful" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case becoming very successful = going great guns.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.