You must unlock four additional horse stable slots at any stable in order to be able to purchase the additional horses necessary for the achievement - as you start out with a horse that appears to be unkillable.
The four slots will cost a total of $1500 ($100, $150, $500 and $750) and I recommend waiting until you have that sum before purchasing any. It is extremely inconsequential, but the more horses you have, the higher your daily stable fees are, so you can save a tiny bit of money by doing this in bulk. It is possible to purchase slots with gold bars as well, but given the slow rate of earning for those, I wouldn't recommend relying on them.
Once you have the slots, the cheapest way to fill them is with Kentucky Saddlers - $50 each for a total of $200 extra. Once you name the fourth horse and it is stabled, the achievement should pop.