Teacher briefly trained house's first champion runner (7)
I believe the answer is:
cheetah
'champion runner' is the definition.
(fastest land animal)
'teacher briefly trained house's first' is the wordplay.
'briefly' means to remove the last letter.
'trained' is an anagram indicator (to train can mean to grow something into a shape).
'house's first' becomes 'h' (1st letter of 'house').
'teacher' with its last letter taken away is 'teache'.
'teache' anagrammed gives 'cheeta'.
'cheeta'+'h'='CHEETAH'
(Other definitions for cheetah that I've seen before include "Spotted creature" , "Fast predator" , "Cat renowned for speed" , "Very fast mammal" , "Member of cat family" .)