The most common and most likely answer to this crossword clue is the 9 plus 6 letter word CHARACTER SKETCH
I've seen this crossword clue in The Sunday Telegraph
C
H
A
R
A
C
T
E
R
S
K
E
T
C
H
More clues leading to the result CHARACTER SKETCH
Buddy explains!
The straight part
We know that "personal description" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case personal description = character sketch.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.