Hi Llawliet,
andykn101 has the removal and reinstall right there. There are some more details though in HOWTO45394. You can do a manual push or a scheduled push to put the client back on the pc that both work in addition to the pull method.
If you're wanting to fully automate the process and want an instant reinstall rather than wait on your scheduled push you could use the ASDK. The resource management service is the one that has the correct method (altiris/ASDK.NS/ResourceManagementService.asmx), otherwise you can also use the community(so far rsteenwyk and myself but more are welcome to contribute!) PowerShell module for the ASDK (Connect Article, Github Repo) to push it using the following command:
Install-AltirisAgentToComputers -Server "NSServer01" -computerNames "ComputerName1,ComputerName2" -showInStartMenu $False -showInSysTray $true -useProxy $False -showInAddRemove $True -overrideInstallPath ""