What you want is PowerShell 3 and its Invoke-RestMethod, ConvertTo-Json, and ConvertFrom-Json cmdlets. Your code will end up looking like:
$stuff = Invoke-RestMethod -Uri $url -Method Get;
and there shouldn’t even be a need to invoke ConvertFrom-Json on the resulting $stuff => it’s already in a usable non-string format.
See http://technet.microsoft.com/en-us/Library/hh849971.aspx for details.