Raspberry Pi SPI interface with PIC-16f877A
The Raspberry Pi currently only implements the master mode of SPI so it cannot function as a slave unless you wanted to create a software implementation of the SPI protocol. Another option would be to create a protocol where the Pi queries the PIC (as master) and get's data back. Per other comments though it is probably easier to implement the communication via serial. 

Finally, the pins on the Pi are not 5V tolerant. The operating voltage of that PIC is 2-5.5v so just be careful if you're powering the PIC with anything above 3.3v.