In a very simplified, very elementary description: Black holes (AFAIK) suck in huge quantities of light and matter and compresses matter into a singularity. The big bang (AFAIK) was the start of our universe through which huge quantities of energy and matter exploded from a singularity. I can't help but make the connection. It seems like black holes are the "opposite" of the big bang, with my very layman understanding of them. People often talk about what happens "on the other side" of the black hole, what if the other side of a black hole is another big bang?
Black holes and the big bang have absolutely nothing to do with each other. The fact that both have the term "singularity" involved in their description is irrelevant, as "singularity" in both cases just means "a place where our math models break down and we don't know WHAT is going on", NOT that the same thing is going on in both cases. EDIT: and by the way, this discussion has happened here about 7,000 times, so a forum search will give you lots of information.
There is a connection of sorts. The 'other side' of a black hole might be [theoretically] a white hole with the characteristics you describe. That's a classic GR view. So far I don't think we have any way to confirm or deny such white holes. In another view, the discrete geometry effects of Loop Quantum Gravity creates a repulsive force which is negligible at low space-time curvature but rises very rapidly in the Planck regime, overwhelming classical gravitational attraction and thereby resolving singularities of general relativity. Whether this is might be a unique big bang or an aspect of cyclic [repeated] bangs remains an area of study. Check out an introduction here if interested: Loop quantum Cosmology http://en.wikipedia.org/wiki/Loop_quantum_cosmology
edit: I see pHinds replied with a somehwhat different view while I was composing above. Another perspective comes from Roger Penrose. He has some mathematical ideas about BB and BH. One interesting relationship, maybe a dichotomy, about the two is that the BB seems to have incredibly low entropy while BH have extremely high entropy. Penrose attributes this to very low gravitational entropy and very high, respectively. Another interesting relationship is that it could be the big bang is a 'singularity' in our past while BH are singularities in our future. Our universe might start with a bang and end as an evaporating series of BH. For more, check out Conformal Cyclic Cosmology in these forums or Wikipedia.
I know one thing they have in common: they are both very elusive to our attempts to understand them. They're both sort of "hidden" from us by mathematical limitations or other sorts, perhaps in some similar ways as dark energy and matter. They all four seem to have this air of "different from the rest of the observable physical Universe" about them. Do they not?
The maximally extended Schwarzschild solution to the Einstein field equation is a solution that contains both a black hole and a white hole, but connected the other way around from what you wrote, i.e., there are future-directed timelike worldlines that run from the white hole to the black hole. This is ruled out on theoretical physical grounds, though. A maximally extended Kerr spacetime (rotating black hole) has blocks that repeat indefinitely, but theoretical work of Poisson and Israel indicates that these tunnels get clogged up. There is also Poplawski's controversial designer spacetime constructed by surgical methods: take two spacetimes, one a white hole, and the other a black hole, use a surgical knife to cut away and toss parts of each spacetime, and stitch together what remains, thus forming a single spacetme. I am not sure this is classical GR, as it requires exotic radiation (negative energy density) on the lightlike hypersurface of stitching.
The search for primordial black holes has been fruitless to date. Only high mass candidates [~10^20 gm] remain viable, at least so far as dark matter candidates. There is no obvious reason why PBH's should have such a high mass. In theory, primordial density fluctuations should generate a wide variety of PBH masses - from as low as 10^-5 gm to over a solar mass, with most at smaller masses.