The TurboTax desktop software only supports a Windows or Mac operating system. 

You can use the TurboTax online editions.

How do you use the Turbotax online for the Chromebook?

You just set up an online account using the browser.  You log into the internet version.  You can start an Online return here

https://turbotax.intuit.com/personal-taxes/online/ 

Not this year.  Had to do my taxes on my windows computer.  Also can get on line using my android. You say I need to update my Chrombook's Chrome browser.  Chrome says I have the very latest level of Chrome Browser . I can get on using my Android and on my Windows computer, but not on Chromebook.  It likes my Username and Password, but WON"T work with my Chromebook.