This is pasted from a post I made here years ago. If all you need is to have the pads play the drum rack sample cells, this might help.
(If you want to have the MPC+Live properly connected where the MPC could start/stop Ableton's sequencer, MPC sequencing things in Live, Or vice versa...check the Akai site. They have a real quick walk through to set that up right.)
http://www.akaipro.com/syncabletonAnd Tutor's book on Ableton+MPC usage is certainly worth getting if you want to easily see all the possibilities of these two working in tandem.
But for just making the pad presses trigger the drum rack -
Select "MIDI From" as either "All Ins" (default) or choose the MIDI port/interface the MPC is connected to specifically (to make the MIDI you send to that channel exclusive and not trigger other instruments on other channels). Press the Rec Monitor button at the bottom of the MIDI channel that the Drum Rack is on.
Drum Racks start with MIDI note # 60 on the first pad, but to the left of the 16 pad grid there is a box that allows you select a group of 16 notes over 7 octaves. A MIDI controller will control whatever box of 16 are highlighted if it's pads are programmed like this
Pad 1- Note # 60
Pad 2 - Note # 62
Pad 3 - Note # 64
Pad 4 - Note # 65
Pad 5 - Note # 67
Pad 6 - Note # 69
Pad 7 - Note # 71
Pad 8 - Note # 72
Pad 9 - Note # 60 (On MIDI Channel 2A)
Pad 10 - Note # 62 (On MIDI 2A)
Pad 11 - Note # 64 (On MIDI 2A)
Pad 12 - Note # 65 (On MIDI 2A)
Pad 13 - Note # 67 (On MIDI 2A)
Pad 14 - Note # 69 (On MIDI 2A)
Pad 15 - Note # 71 (On MIDI 2A)
Pad 16 - Note # 72 (On MIDI 2A)
The square highlight box on the right side of the drum rack with the 128 squares can also be used to target what pads will be triggered. I believe the pads can just be in chromatic order from Note #36 and up and it will trigger the pads, if you don't want to manually make the MIDI Note # assignments listed above.
BTW-
To make a pad press trigger the whole sample, highlight the pad and press the third circular icon on the far left of the Drum Rack. Now you have a waveform view and controls for each sample/pad. Set the release on the ADSR envelope controls to maximum.