There should not be a "bootstrap class". It is a straightforward process, which can be contain in a simple script, which would serve as entry point for your application. PHP is not Java, therefore you are not require to contain everything within a class.
Usually bootstrap stage of application would have following responsibilities:
- set up the autoloader
- initialize routing mechanism
- configure storage abstractions (db, cache, etc.)
- handle the user request (using the routing)
- dispatch to MVC
The bootstrap stage in you application is where all the wiring between objects should be set up. It would also be the place where you set up such things as loggers, access control and error handling structures.
You could say that front controller is a part or bootstrapping.
P.S.: also, you might find this answer of mine relevant, since it also contains an example of bootstrap file.
List of recommended articles:
The last two links cover two of 3 major MVC-inspired patterns (Model2 MVC and MVP), since classical MVC is actually highly impractical (and actually, almost impossible) to be used for web applications.