It looks like that is a motor + geared screw drive actuator. If so, you need to apply power with one polarity to push the ram out and reverse the connections to retract it. It will not retract on its own when you remove the power nor will the weight of the load force it back in (unless severely overloaded - it would probably break first). It claims to have internal limit switches which would disconnect the power at the ends of its stroke. The usual arrangement has a diode in parallel to each limit switch so it can be reversed away from the limit. However, don't trust them - if something jams it and it doesn't reach the end of its stroke, it will burn out if you don't cut the power to it in under two minutes, so its best to time how long it needs end to end moving your load, then turn the relays off after double that time.
Your proposed connections wont work - or at least will only work once. You need a (relay) H-bridge to reverse the power to it as required. Connect 0V to the NC terminals of two of the relays and 12V to their NO terminals. Don't forget a fuse in the 12V feed, 1A should be suitable. Connect the actuator between the COM contacts of the two relays. With both relays off there is no power to it. Turn either relay on and it will go in or out depending on which relay and which way round you wired it. If you turn both relays on, it will put 12V on both of its wires, no current will flow and nothing will happen, so you don't have to worry if that happens by mistake.