If the target creature is an illegal target by the time Blood Aspirant's last ability tries to resolve, the ability won't resolve. The creature can block as normal. If the target is legal but not dealt damage (most likely because of a prevention effect), it can't block this turn.

Blood Aspirant's first ability is a triggered ability, not an activated ability. It doesn't allow you to sacrifice a permanent whenever you want; rather, you need some other way of sacrificing permanents, such as its second ability.

If you sacrifice a permanent as part of casting a spell or activating an ability, Blood Aspirant's first ability will resolve before that spell or ability.

Blood Aspirant can be sacrificed to pay for the cost of its own last ability.