I built one of these pulse rate sensors for a company whilst at university back in 1971. Using similar components that were then available - but the solution didn't include any processors!!
Key things to think about. The level of the signal you are interested in is 'swamped' by the level of "DC" voltage seen by the sensor. This "DC" offset voltage will vary enormously between people and also for the same person in different conditions (e.g. hot or cold, well or ill).
You should shield the sensor from external light - especially mains driven - which adds a further signal conditioning problem - that's why the ones used in (eg) hospitals clip over the finger. With care you should be able to get some readings but I suspect that the variations in output levels will make it difficult to detect accurately.