Beginnings of cool line in some computer images (4,3)
I believe the answer is:
clip art
'computer images' is the definition.
(I've seen this before)
'beginnings of cool line in some' is the wordplay.
'beginnings of' indicates taking the first letters.
'some' becomes 'part' (**).
The initial letters of 'cool line in' is 'cli'.
'cli'+'part'='CLIP ART'
(Other definitions for clip art that I've seen before include "Widely used images" , "Pre-drawn pictures usable in computer packages" , "computer images" , "Graphics files" .)