As mentioned in the comments, there are ways to ensure that you can get unlimited souls.
Assuming that you are talking exclusively about offline play (playing online would also allow for a major portion of souls - until you surpassed everyone in your soul memory range) there are many ways to increase your soul collection. Items such as the Silver Serpent Ring, and the Tseldora Armour Set, can increase the souls you receive, making it easier to farm souls.
However, through the use of Bonfire Ascetic farming (as user Question Marks already suggested), this allows for unlimited soul farming. Additionally, the Stone Soldiers in Drangleic Castle respawn indefinitely, as they are needed to open the Soul-devouring doors. Even without employing any other methods, this is still possible to gain the max amount of souls in one playthrough.
As for what the cap actually is for souls held, well, the best I could find was to do with Soul Memory. The top tier extends up to 999,999,999 souls; so either that is the point where they stop counting, or they just cap soul collection there, since even levelling up from level 837 to 838 only takes a couple million souls (1.5-2 million; somewhere in that range, I believe).