Update JSON file using PowerShell

Your json is missing the starting and ending curly brackets:

{
    "policies": {
         "Framework.DataContext": {
            "connectionString": "Server=ServerName;Database=DateBaseName;Integrated Security=sspi;"
         }
    }
}

Now you can update the file like this:

$pathToJson = "F:\Path\To\JSON\file.json"
$a = Get-Content $pathToJson | ConvertFrom-Json
$a.policies.'Framework.DataContext'.connectionString = "Server=ServerName;Database=DateBaseName;Integrated Security=sspi2;"
$a | ConvertTo-Json | set-content $pathToJson

You could also use some Select-Object to get the property:

$connectionString = $a | select -expand policies | select -expand Framework.DataContext 
$connectionString.connectionString = 'Test'

$s = Get-Content "F:\Path\To\JSON\file.json" -Raw|ConvertFrom-Json
$s.policies.'Framework.DataContext'.connectionString="Server=ServerName;Database=DateBaseName;Integrated Security=sspi2;"
$s|ConvertTo-Json |Set-Content "F:\Path\To\JSON\file.json"