How do you unlock REX?
There are several steps to unlocking REX.
In the first level, check the back of your escape pod. You should be able to take the Fuel Array. This item will be placed in your equipment slot. NOTE: Because MUL-T does not have an escape pod, that character is unable to obtain the Fuel Array, which in turn means it can't unlock Rex. Use another character!
Carry the Fuel Array through several levels. This is somewhat difficult because not only does it take up your equipment slot, it will also detonate (which kills you) if you fall below 50% health. The sacrificial shrines that take your hit point are basically a guaranteed suicide while holding this item. Other enemies with high burst damage (golems, Wandering Vagrant, etc) also become much bigger threats. And of course, manually activating it will also detonate it, so stay alert!
Bring the Fuel Array all the way to the Abyssal Depths (stage 4). Once there, use the geysers to propel yourself to the highest platform. There should be another platform right behind it, semi-hidden near a giant rock column. On that platform is the unpowered REX. Go over to him and activate him, which will remove the Fuel Array from your equipment slot.
Congratulations! This will give you the Power Plant achievement, for which the reward is none other than REX himself.