How to properly use the -verbose and -debug parameters in a custom cmdlet

$PSBoundParameters isn't what you're looking for. The use of the [CmdletBinding()] attribute allows the usage of $PSCmdlet within your script, in addition to providing a Verbose flag. It is in fact this same Verbose that you're supposed to use.

Through [CmdletBinding()], you can access the bound parameters through $PSCmdlet.MyInvocation.BoundParameters. Here's a function that uses CmdletBinding and simply enters a nested prompt immediately in order examine the variables available inside the function scope.

PS D:\> function hi { [CmdletBinding()]param([string] $Salutation) $host.EnterNestedPrompt() }; hi -Salutation Yo -Verbose

PS D:\>>> $PSBoundParameters

____________________________________________________________________________________________________
PS D:\>>> $PSCmdlet.MyInvocation.BoundParameters

Key Value                                                                                                                                                                                                           
--- -----                                                                                                                                                                                                           
Salutation Yo                                                                                                                                                                                                              
Verbose   True                                                                                       

So in your example, you would want the following:

function DoStuff `
{
    [CmdletBinding()]
    param ()
    process
    {
      new-item Test -type Directory `
        -Verbose:($PSCmdlet.MyInvocation.BoundParameters["Verbose"].IsPresent -eq $true)
    }
}

This covers -Verbose, -Verbose:$false, -Verbose:$true, and the case where the switch is not present at all.

Perhaps it sounds strange, but there isn't any easy way for a cmdlet to know its verbose or debug mode. Take a look at the related question:

How does a cmdlet know when it really should call WriteVerbose()?

One not perfect, but practically reasonable, option is to introduce your own cmdlet parameters (for example, $MyVerbose and $MyDebug) and use them in the code explicitly:

function DoStuff {
    [CmdletBinding()]
    param
    (
        # Unfortunately, we cannot use Verbose name with CmdletBinding
        [switch]$MyVerbose
    )

    process {

        if ($MyVerbose) {
            # Do verbose stuff
        }

        # Pass $MyVerbose in the cmdlet explicitly
        New-Item Test -Type Directory -Verbose:$MyVerbose
    }
}

DoStuff -MyVerbose

UPDATE

When we need only a switch (not, say, a verbosity level value) then the approach with $PSBoundParameters is perhaps better than proposed in the first part of this answer (with extra parameters):

function DoStuff {
    [CmdletBinding()]
    param()

    process {
        if ($PSBoundParameters['Verbose']) {
            # Do verbose stuff
        }

        New-Item Test -Type Directory -Verbose:($PSBoundParameters['Verbose'] -eq $true)
    }
}

DoStuff -Verbose

It's all not perfect anyway. If there are better solutions then I would really like to know them myself.

There is no need. PowerShell already does this as the code below proves.

function f { [cmdletbinding()]Param()    
    "f is called"
    Write-Debug Debug
    Write-Verbose Verbose
}
function g { [cmdletbinding()]Param() 
    "g is called"
    f 
}
g -Debug -Verbose

The output is

g is called
f is called
DEBUG: Debug
VERBOSE: Verbose

It is not done as direct as passing -Debug to the next cmdlet though. It is done through the $DebugPreference and $VerbrosePreference variables. Write-Debug and Write-Verbose act like you would expect, but if you want to do something different with debug or verbose you can read here how to check for yourself.

Here's my solution:

function DoStuff {
    [CmdletBinding()]
    param ()

    BEGIN
    {
        $CMDOUT = @{
            Verbose = If ($PSBoundParameters.Verbose -eq $true) { $true } else { $false };
            Debug = If ($PSBoundParameters.Debug -eq $true) { $true } else { $false }
        }

    } # BEGIN ENDS

    PROCESS
    {
        New-Item Example -ItemType Directory @CMDOUT
    } # PROCESS ENDS

    END
    {

    } #END ENDS
}

What this does different from the other examples is that it will repsect "-Verbose:$false" or "-Debug:$false". It will only set -Verbose/-Debug to $true if you use the following:

DoStuff -Verbose
DoStuff -Verbose:$true
DoStuff -Debug
DoStuff -Debug:$true