You could use the Clipboard class from the System.Windows.Forms namespace:
Add-Type -Assembly System.Windows.Forms | Out-Null
$clp = [Windows.Forms.Clipboard]::GetText() -split "`r`n"
The clipboard content is returned as a single string, so you need to split it at line breaks to get an array of lines.
Another option would be the InternetExplorer.Application COM object:
$ie = New-Object -COM 'InternetExplorer.Application'
$ie.Navigate("about:blank")
while ($ie.ReadyState -ne 4) { Start-Sleep -Milliseconds 100 }
$clp = $ie.Document.parentWindow.clipboardData.getData('text') -split "`r`n"
$ie.Quit()
However, for this to work you must put about:blank in a security zone where you allow programmatic clipboard access for scripts:
