Press alt + tab to minimise the game, open the Teamspeak application, then click back on ESO when you're ready. The game will be in the same place you left it provided you don't reach the inactivity timer (10-15 mins, something like that).
There's no way to open Teamspeak through the game itself as far as I know, you have to tab out each time you want to use Teamspeak - unless you have Overwolf installed as well, in which case you can see Teamspeak over the top of your game as you play (who is in your channel, who is currently speaking, typed messages, etc.)
Hope that helps!
PC NA & EU || Mammoth Guilds - Victory or Valhalla || Altmer sorcerer main
"Wood Elves aren't made of wood. Sea Elves aren't made of water. M'aiq still wonders about High Elves."
Another thing you can do is get a cheap second monitor. I have my TeamSpeak moved to that monitor, so I hit that Windows key on my keyboard, start TS, and it comes up on my second monitor. This way, I don't have to have it on my game screen, and I can see who is talking. This may not be an ideal solution for all since it involves spending money, but it really is nice to have that second monitor.