To drink to Hamlet’s better breath is to drink to something slightly ambigious. On one hand, it could suggest that Hamlet’s breath, his power, his life, his vigor is better than Laertes! However, it could also suggest that Hamlet’s breath needs to be better, that it could use improvement, that it needs support.
And underlying all of it is the knowledge that Claudius wants to STOP Hamlet’s breath, to end his life.
Is there breath in the after life? Do angel’s have breath? If so, it is probably better than earthly breath – so drinking to better breath might be drinking to a kind of a heavenly one.