Yes you can control the motor by pwm but you need some motor driver that can deliver enough current. If you want to reverse the direction you will have to use a h-bridge. If you just need on/off control you can do with 2 dpst relays.
Converting a motors rotary motion into a linear one can be more complicated than you think depending on what precision you need. If you post a sketch of what you are trying to do it would be easier to help you
If you didn't already, you'll probably want to consider limit switches so the motor doesn't drive for ever against the end stops or alternatively power the platform off the rails so to speak.
This page suggests micro switches or Hall Effect Sensor which would need a magnet on the platform; you could also use a reed switch instead of the HES.
you can make one, or buy one. it is actually cheaper to buy one.
you can use a relay as the top of an h-bridge on a single motor.
the relay could send power to one side of the motor or the other.
then us a transistor as the lower half on each side. then pwm the transistor.
any cheap switch can be used as and end switch.
even a pair of LED's can be used as an optical interrupter.
one LED shining into another will have the second output a value. small, but there.
you could have a couple LED's, one top, one bottom and an light sensor on the elevator, when it sees the lights, it knows it is at the top or bottom.