I always thought that it would be a case of continuing as normal, but wasn't sure about ATC procedures etc for the incident.
If you don't have any problems with the aircraft then - and this isn't meant to be flippant - in most cases ATC don't much care. I've never worked an airport that had a procedure specifically for a birdstrike on approach. If you report it the response will vary depending on the controller's assessment of the situation. If you report it you may well be asked if you want to declare an emergency, and it's your call - if you think you may have any sort of problem then do so and you'll probably be met by a fire vehicle or two...but that's what they're there for.
If you have a strike on or over the runway then do report it - even if there's no other problem - so that the runway can be inspected and any remains removed. As an aside, one of the main reasons that remains are removed is so as not to leave a tasty looking meal for scavengers, which are often bigger than the bird that came off worse from the strike. Also, of course, the inspection will check for any bits that may have been broken off your aeroplane - something that can't be done if the strike happens on the approach.
From the inside, my experience suggests that it can be very noisy (but in the case I recall it was a PA38 and they rattle about and creak quite a bit without the aid of a bird!) - and a shock if you don't see the bird. From the size of the dent it left though I guess it had to have been a fairly big beggar.