This achievement requires the player to deliver just one freight of heavy cargo. It can be damaged, you may also be late, as long as you just succesfully deliver the cargo to its destination.
These jobs require a truck that's capable of hauling such heavy loads. Taking a quick job will always result in trucks that can handle it, but your own truck may need to be modified. The best truck for these jobs imho seems to be the Volvo FH16 Classic. Fully upgraded, that thing's a beast.
In the job screen you can add filters. If you choose the filter on the weight you'll probably get the heavy cargo jobs first.