And, like the haggard, cheque at every feather
That comes before his eye.
A "haggard" is a hawk captured when it is an adult. Such hawks are hard to manage, and likely to suddenly go off mission and "cheque" at targets of opportunity, or even at a false target, such as a feather. In comparing the Clown's foolery to a haggard, Viola is making the point that the Clown has to be constantly alert to any opportunity to make a jest that will amuse those for whom he performs.