4-20mA is an industry standard signal, the purpose of that relay would be something like a high alarm (a sensor reading over ~75%)
as for driving it, if you have access to both ends, or even just the bottom end, then a single op amp current sink (look at daves dummy load) will fit the bill, even using that 250 ohm resistor as the measuring resistor if you want to keep the 1-5V signal logic
Just please search this site a bit, there are 20 odd threads showing the small tweaks that had to be done to make them stable (Dave oversimplified the schematic) which also include working component combinations,