I post this with this caveat up front: there are always exceptions to the rule.
But with that said, I was once told by a very influential individual that you need approx. 6 publications to be competitive for one of the top predoctoral internships. So the 3-4 estimate for a tenure track position is obviously low.
I don't know if there's a hard and fast rule about # of pubs, but obviously you'll want:
1. First-authored pubs in reputable journals, and a strong record of collaboration.
2. A coherent program of research - this will be especially important for the job talk. You'll want to present what you've done, why you've done it, and what you plan to do in the future. In other words, it has to flow and make some sense in terms of its trajectory. Having translational research, or a research focus that is "hot" will certainly help.
3. A record of extramural grant funding. This is often hard to do as a grad student/intern/postdoc, but there are mechanisms that you can pursue. For example, the NSF or NIH F31 grants will fund predoctoral research. Sometimes there's a request for applications (RFA) for dissertation research via the R03 mechanism. The NIH F32 will fund postodoctoral research, and when you're a postdoc you'll want to write at least one grant (i.e., R21, R34, R03, or K award). Finally, there are non-profit orgs that sometimes fund research (e.g., NARSAD). Basically, research I universities want to see that you can bring in the big bucks.
4. Depending upon the school, teaching record may or may not be important.
Finally, working with reputable people in your field will be important. Networking is key - psychology is a small community.
I think it's great to understand how these things work up front, but it's also important not to freak out too much about this stuff now. Things often fall into place as you go through graduate school.
Good luck!
ETA: I didn't directly answer your postdoc question b/c I think it's mainly a question of where you stand on the above items. If you're ready to go after internship, it can be done. But if you need to beef up in any of these areas, a postdoc is ideal because it provides the opportunity to do so.