Pressing backspace with root gives weired character in putty

You can reset the key while logged in by entering stty erase <then hit the backspace> That will set the backspace to whatever character you are sending as backspace for that terminal session. tylerl's response is a better long-term fix.

In the putty options under Terminal >> Keyboard there's an option to change your backspace emulation. Change it and see if it makes things better.

Alternately (or additionally) mess around with the checkboxes under Terminal >> Features to see if one of those (such as "Disable destructive backspace...") fixes your problem.

There's a complex interaction between terminals and hosts using a set of conflicting standards which makes getting the right match a bit of a trial-and-error exercise.

And just something to have in mind. Generally, if backspace doesn't work, try with ctrl + H

If the backspace was working okay initially and no longer is then try using the "reset" command. Look around putty's options menu for a "send hard reset" option as well. If it was never working then try what tylerl suggested.