How can i perform this move?

There are two methods for eliminating a player during the royal rumble:

Method 1:
Start within "arms reach" of the rope. Irish whip (R1 + tap X) victim at the nearby rope. Too far and he'll bounce off the ropes and run back at you... holding R1 + X leads to the strong Irish whip / roll him out - pull him back move.
Once victim flies over the rope, use right analog stick to initiate a grapple. This starts a "ring out minigame" that works just like a submission move (mash buttons and spin move stick around). You should remove 2/3 of his "ring out bar" if solo - if victim has less than 2/3 ring out bar when the move is started, he should be eliminated.

Method 1.5 - after a victim is thrown onto the apron, a second wrestler can approach the fight and lend his strength to the ring-out battle. 99% of the time, the victim cannot withstand 2 people working on the throw-out, and he'll be eliminated.

NOTE - I find it best to store a finisher, when facing a ring-out situation of your own, executing the finisher (L1) will get you right back into the ring, and, if done before someone starts the ring-out process, it saves your ring-out bar.

Method 2 - Requires the possession of an ultimate control move that can lead to a throw. IIRC, piledriver and whatever move is the military press are two of the best UCMs for the rumble, as they both have forward throw animations.
To use this method, you need to get the opponent to have at least 1 red limb. Then remember which direction has the appropriate UCM mapped to it - on my CAW, piledriver is mapped to the "option 2 - right" grapple moveset, and the military press is mapped to the "clean - up" grapple moveset. I prefer to initiate the right grapple moveset (piledriver), so I use [R1 - right analog stick - right] to start this sequence. In the grapple, press [R3] to initiate the UCM. Once the menu comes up, [Right analog stick - up] lifts the victim over your shoulder with his arms spread out in an airplane-esque shape. Quickly walk to the nearest rope, face that rope, then press [Right analog stick - down] to toss him right over the rope and onto the floor for an easy elimination. NOTE - using the UCM while the opponent does not have a red limb will be ineffective, as you will slide back and the target will land just inside the ring against the rope...
For the record, the military press one is the one where you lift the guy up barbell like over your head when you press R3, and repeated up-down sequences on the right analog stick has him being lifted like a barbell, but going straight to down or second pressing of R3 (can't recall which right off top of head) leads to victim being tossed "over the shoulder" in a variant of Cena's attitude adjustment / fireman's carry. When using this move, keep him close to the head (to walk), lift up at rope, then hit R3 when he's up in the air for a forward toss right over the rope.