Is there a center of gravity in black hole? I've come across this particular gif 
"http://i.imgur.com/AOCqg5j.gifv"

If you can see the above gif, you can see the two perfectly spherical black holes merge together forming a single larger sphere. My understanding of celestial bodies forming a spherical shape is because gravity is "round." 
But my understanding of black holes is the object's density is infinite (and uniform?). In other words, how can we speculate that black holes will follow conventional "center of gravity" and thus form spherical shapes? 
 A: Some misunderstandings. 
Black hole can be round (we call them spherically symmetric, or spherical), or sort of ellipsoidal, or axially symmetric, if they have angular momentum (i.e., if they rotate). That is true for stationary black holes, i.e., after they achieve a stationary state. In getting to that state they can be very dynamic and have deformed (from the final state) shapes, like you see in the simulation gif you posted while they are merging and settling to the final state. 
There are well known theorems in General Relativity (GR), stationary black holes can only be one of those two, plus an additional possibly that they also have charge, but the two shapes are as stated. Because there are only three parameters that define stationary black holes, mass M, angular momentum J, and charge Q. It is said that that's the only hair black holes can have (i.e., those are the only three things you can detect form outside). The theorem is commonly mentioned as 'black holes have no hair', well with those 3 exceptions. 
Most black holes have some rotation and no charge (charges will get cancelled by opposite charges flowing in, typically). The GR solution for that is called the Kerr solution or black hole. The spherical one the Scharzchild solution. 
There is NO way to compute the center of gravity, all the mass in both cases, is presumed to have fallen into the singularity. For spherical symmetry that's a point in the geometrical center,  for rotating black holes it is an extended shape (sort of circular or ellipsoidal, really I forget) along the equator inside surrounding the geometrical center. We can NOT see any of those singularities, they are inside the event horizons. Center of gravity only makes sense as to we'd measure the distance to the black hole, ideally it's the geometrical center, but we are a heck of a long time away from being that accurate. The distance to the merged black holes announced earlier in 2016 was estimated at about 1.3 billion light years, with accuracies of a few percent (again forget exactly but think maybe 5 to 10 percent). A few tens or hundred meters to an effective center of gravity is not measurable anytime soon. 
A: Yes, they do have a center of gravity. First, you must comprehend the fact that black holes are not point objects. They do have finite dimensions. Any mass M can be turned to a black hole by shrinking it to a radius equal to schwarzchild's radius 

To put that into perspective, you can turn earth into a black hole by shrinking it to the size of a peanut. So, now, you can treat them as two spheres of finite radius and find their center of gravity.
A: Black holes, as seen in the picture are actually spheres formed by event horizon. The matter is all concentrated on the singularities (except for the matter that is falling into either singularity at a given point in time). So, individual black hole would be spherically round. During merger of two black holes, the event horizon can become non-round intermittently, turning into a sphere again post merger. The uniform density question does not make sense as the size of singularity is not well defined.
