When my opponent declares blockers, I want to be able to kill a blocking creature (with something like Hero's Downfall) while still doing combat damage to my opponent.
Is this possible?
The last time you can cast an instant to prevent a creature from blocking is in the Assign Attackers step, after attackers have been declared. If you have moved to the Declare Blockers step, it's too late, since the very first thing that happens is the defender declares blockers. Nobody has priority until after that happens.
To answer the question in your title, yes, each player receives priority and can cast instants after blockers are declared. However, to answer the (unwritten) question in the body of your text, the attacking creature will still be considered blocked, even if the defender is killed or otherwise removed from combat. A blocked creature will deal no damage to the defending player unless it has trample (or some other card-specific ability).
It sounds like you want to kill an opponent's creature before blockers are declared. Your last chance to do this is when you receive priority after attackers are declared. If you kill potential blockers at this point, they won't be around to be declared blockers, meaning the damage will get through to the opponent.
You can't respond to blockers being declared the way you could respond to a player casting an Instant, as declaring blockers is a turn-based action that doesn't use the stack.
This means that once the game moves toward the Declare Blockers Step, your opponent will get to declare blockers, all attacking creatures that were assigned a blocker become blocked, and then players get priority. Even if you destroy a creature using Hero's Downfall at this point, the creature it was blocking is still blocked and generally won't deal combat damage.