painting fabric, even with fabric paint, will make the fabric stiff.

what you want to do is dye it, not paint it.

if you want to make it black, i would get some RIT dye, and test it on a scrap of similar fabric. the same would be best, but if you can't get a swatch, the same type of fabric will do.
try the dye full strength, just paint it on a corner. then do another corner with half strength, another with 1/4 strength, and one with 1/8 strength. after it's dried, see which you like best, and if it changes the fabric texture at all. it shouldn't, but it's good to check.

once you know the strength you want to use, go ahead and 'paint' the ottoman with it. use a cheap throw away brush. a spray bottle might work, but your hand will get mighty tired from all the pumping.

make sure you do it on plastic drop cloths, so that you don't drip on anything. and plastic bag+tape off any wood feet that you don't want dye on.

make sure to let it dry a day or 2. more humidity = longer drying time.

to heat set the color, you'll need a super hot iron and a pressing cloth of some variety. you'll put the iron on the highest setting it has, usually wool (don't use steam!), and with the pressing cloth between the iron and the fabric, rub in circles for about half a minute. let it cool down. you can do this more than once if you find that rubbing the fabric transfers dye. leave it a couple hours before testing the rub, so that the dye can cool and stabilize.

sounds complicated, but it's really not hard to do at all.
the hardest part would be the heat setting.