Extremely dodgy and disgusting line (7)
I believe the answer is:
dynasty
'line' is the definition.
(I know that dynasty is a type of family)
'extremely dodgy and disgusting' is the wordplay.
'extremely' says to hollow out the word (remove centre letters) (only the extremes of the word are used).
'and' says to put letters next to each other.
'disgusting' becomes 'nasty' (disgusting-tasting).
'dodgy' with its middle removed is 'dy'.
'dy'+'nasty'='DYNASTY'
(Other definitions for dynasty that I've seen before include "Line of kings" , "Sequence of hereditary rulers" , "Succession of rulers from one family" , "Family of successive rulers" , "Powerful family line" .)