“you did painting need, And therefore to your fair no painting set”
'Painting' here means artificial embellishment or rhetorical decoration applied to a subject to make it appear finer than it is. The speaker says he saw no need to add such ornament to the youth's beauty because the beauty was already sufficient — to paint it would have been to imply it was deficient.
historical The blazon (and the anti-blazon)“barren tender of a poet's debt”
'Tender' is a legal term meaning a formal offer of payment of what is owed. A poet owes praise to a patron who inspires him; that praise-tribute is the 'poet's debt.' 'Barren' means unfruitful, producing nothing of value. The whole phrase means: the meager, worthless payment a poet makes when he merely goes through the motions of praising someone who has surpassed all praise.
“I slept in your report, That you yourself, being extant”
'Slept in your report' means the speaker let the account of the youth's qualities lie dormant — he did not write about them, keeping the record asleep rather than active. 'Extant' means still alive and present in the world. The argument is that since the youth himself is still living, his own presence demonstrates his worth more eloquently than any written report could.
“a modern quill doth come too short”
'Modern' in Shakespeare regularly means 'ordinary,' 'commonplace,' or 'everyday' — not 'up-to-date' as it does today. A 'modern quill' is a run-of-the-mill contemporary writer's pen, not a brilliant or exceptional one. The phrase concedes that even the best ordinary poet falls short of capturing the youth's worth — and implicitly includes the speaker among those ordinary writers.
historical The Fair Youth, the Dark Lady, and the Rival Poet