Supposedly he did everything including framing Bucky, so that all the Avengers would come to a place where he can make Stark watch the video of his parents death in the hope that would make him turn on the rest of the Avengers.
I think his primary objective throughout the movie was to obtain hard evidence that the Winter Soldier has killed Stark's parents, in order to turn Stark against the Winter Soldier and presumably Captain America. I don't think he necessarily originally planned to have them meet him in Siberia.
However, as events played out, he eventually decided that dropping the false trail of the inactive Winter Soldiers would likely lead the Avengers there.
But how could he know that Stark (Iron Man) would even show up in that place?
Rogers and Barnes arrive and state "[Zemo] got here a few hours before us." They then gear up and enter the base. Moments later Iron-Man arrives and joins them. Then they ride a noisy elevator down into the facility.
Assuming the Avengers took the bait of more Winter Soldiers, he would expect some of them to show up (maybe he isn't current on how many are imprisoned or paralyzed). The elevator opens and it is Barnes, Roger, and Stark, lucky him! At this point he decided to reveal himself and show Stark the tape, to get them fighting. Had they not arrived, he might have left the base and found a way to get the footage to Stark at a later time.
If Stark hadn't been there, he might not have revealed himself (we may never know the truth).
And was Zemo the one who hired Rumlow in the beginning to steal the biological weapon, if so how did that fit in his plan?
I don't think there is anything to suggest that is the case. That sequence of events was more designed to demonstrate that the Avengers cause a lot of collateral damage.