Yes if you have a catalog backup and your images backed up. You can convert it to LR6 or LR CC 2015.
If you still have all the images, and have access to the catalog that you were using with Lightroom 3, then you should be able to upgrade without losing anything. The installation of the upgrade will create an updated copy of your catalog, and all of your adjustments should remain intact. Lightroom 3 used an older process version, but the new version will honor that version and those adjustments for all the work that you have done. New imports will automatically use the new process version. But if you really want to you can switch back to the older one. Personally, I think process version 2012 is much better.
I suggest you download Lightroom CC and use it free for 30 days. You then have the choice to subscribe or to convert the trial into LR6 as a stand-alone product.
computer crashed, but I do back up computer on separate hard drive. New computer - new app, right? thanks for your help
Copy your photos from your old computer backup to the new computer using the same folder structure as on the old computer.
Copy your LR catalog (the .lrcat file) from the old computer to the new computer.
Install LR CC from the link, above.
Open the LR CC app on your new computer.
Use File / Open Catalog to open the copy of the .lrcat file on your new computer and indicate you want to convert to a new LR 6 format.
Decide if you like LR CC or not.
If so, pay for a subscription or pay for a serial number.
For reference, here is how to move things from one computer to another:
Where you’re not able to start up LR on the old computer, so you’d need to start with Step 3 Install LR on new computer.