Q: I can view a persons message on my phone but it will not show on my computer

See "Set up SMS and MMS with Continuity" here:

Use Continuity to connect your iPhone, iPad, iPod touch, and Mac - Apple Support

In order to see iMessages (blue on your phone) on both devices, you need to be logged into the same Apple ID on both devices. In order to be able to see SMS messages on your computer, you need to have text forwarding set up. Settings>Messages>Text Message Forward. You'll also need to have both devices on the same WiFi network.