There are 2 ways to activate the Adamantium bonding sequence. The 1st is to recover the memory by hitting the Sink hole to the bottom right of the Sabretooth Claw spinner 3 times and then landing the ball in the spinner and being lucky enough to land on the Adamantium game. Not something I would put much faith in.
The 2nd is the best way to try and can be done right from the start. You simply need to hit the left ramp 4 times to light up the word BOND. The catch is that if the ramp is lit for a Combo then it won't progress the BOND letters. Simply take your time and rest the ball on the right flipper, make sure that combo isn't flashing over the left ramp which brings you in through the Weapon X tank. Once you have lit BOND you need to put the ball in the left sink hole where it spells out SNIKT and the mode will begin. I don't think there is any time limit on hitting the sink hole so take it slow.
This is where it gets a bit tougher. This is a 2 ball multiball mode and you need to first get one ball into a sink hole. When you do this the table will start to glow green and it's not immediately obvious what you need to do. You have about 6 - 8 seconds while the table glows green in which to get the 2nd ball into the other sink hole. Once you managed to do that one of the balls will become an Adamantium ball and will be glowing white. Simply put this ball through the left ramp (Weapon X tank) and the achievement is yours.
***EDIT*** As it's been mentioned in the comments a couple of times I'll clarify. You need to put the Adamantium ball (glowing white) through the left ramp but you must keep the 2nd ball active. Draining either ball will end the mode and you'll need to start the process again.
It's a pretty difficult achievement to get so I'd recommend taking your time and learning how to hit the sink holes consistently. The sink holes are the key to big points on this table and play a big part in the Sabretooth fight and can get you huge points on the Sentinel mode as well as being needed for the Samurai multiball.
Take the time to learn how to light up the SNIKT from the left sink hole as lighting that activates the claw save which can be invaluable to keeping your game going especially as this can be a fairly unforgiving table.
If any of this in unclear or you want some other tips let me know and I'll happily add more detail.