Can you gain achievements while in Offline Practice Mode?

Besides the one training mode achievement, no, you can't.

Considering you are online, you could create an empty server for yourself and add some bots (tf_bot_add) to it (just make sure sv_cheats = 0). I've gotten a few achievements this way. Also, be sure that you're using a supported bot map, otherwise you'll need to use nav_generate.

Hope this helps.

As far as I know you can only gain achievements online. Plus must of the achievements require you to be online. Really. I have all the TF2 achievements, so I know.