SSL is designed to keep eavesdroppers out, and, from your problem description, you want to be an eavesdropper...
In the handshake, the
Certificate message is from the server, not the client, so you won't learn client details from that message. Of all the client messages, only the
ClientHello can possibly teach you a few things, but this will be limited. Basically, you will have the list of supported cipher suites (with ordering), compression support, and the presence or absence of SSL extensions (e.g. Server Name Indication). Such parameters depend on the browser and the OS, but with only few variations, so don't expect miracles. If the client browser is Firefox, then it has its own SSL library (called NSS), but you won't be able to distinguish between Firefox on an Android smartphone and Firefox on a Windows machine. Otherwise, you may hope for recognizing Apple systems (iPad, iPhone, Mac OS X), Windows and Android/Linux as three different groups, but you won't get more details. If you know the client OS and browser types, then you might be able to work out the approximate version based on the presence or absence of some options.
At least some people have toyed with the idea of collecting SSL
ClientHello information from many browsers to try to work out browser/OS types and versions. E.g. see these pages from Qualys SSL Labs. This still seems to be at the conceptual stage (and not actively developed).