Give running account of one leaving red books at Eton's entrance (10)

I believe the answer is:

commentate

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'give running account of one' is the definition.
I can't judge whether this definition defines the answer.

'red books at eton's entrance' is the wordplay.
'red books' becomes 'comment' (I can't explain this - if you can you should give a lot more credence to this answer).
'entrance' indicates taking the first letters.
The first letter of 'eton' is 'e'.
'comment'+'at'+'e'='COMMENTATE'

'leaving' acts as a link.

Can you help me to learn more?

I've seen this clue in The Independent.

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