If one is using a VPN from the local machine, how much can a hacker on the local networks router inspect, modify or hijack the packets?
That depends on your own VPN configuration. For example, the OpenVPN documentation warns that Man-in-the-Middle attacks are possible, if the certificate is not verified.
When configuring OpenVPN (or any similar VPN software) that way, you essentially tell it "Connect to this server and trust whoever answers on the other side". If an attacker has hijacked your router, it's very well possible that the attacker directs you to an attacker-controlled VPN server, which in turn gives them access to all your connection data.
If, however, your VPN connection is set up to verify that the certificate belongs to who you expect it to be, either through PKI or TOFU, then your VPN software should give you a warning that either no certificate is present, or that the presented certificate does not match what is expected.
In this case, VPNs are not more or less secure than a HTTPS connection made to any HTTPS server (such as https://stackexchange.com). You will receive a warning that no secure connection could be made and you should abort the connection attempt. This will, essentially, leave you stranded without any connection, until you get the attacker off your router.