By far, the easiest way for me to propagate black currents with hardwood cuttings. In 
winter, during dormancy, when I prune black current bushes, I make pensile size cuttings with exactly 5 buds. After my cuttings are ready, I take 5 gallon bucket and make multiple holes on bottom, with a drill bit half an inch or 3/4th. Fill up the bucket with regular compost, potting soil, just whatever is available and stick the cuttings in. I keep 2 buds above the soil and 3 buds below, submerged in compost. Lightly water everything and put it aside to shaded place.
The cuttings will show signs of growth during spring time, and new growth of green wood will appear at the end of summer. I keep an eye on cuttings during the spring and summer month just to keep them watered and never let them dry out. It is very important to keep the soil moistened to allow new roots develop. If wood starts flowering or even sets fruit, I always try to remove it to concentrate all energy to root growth.
At the end of 1 season, about fall or winter time, I have perfect starts of black current plants with several branches and very nice root system developed. All is left, just to transplant them to permanent location or bigger pot. Easy, simple and does not cost me anything!