How can I get tax details from my personal webserver and display on the checkout page in shopify?

You can use an App Proxy at the cart level to show off information. The proxy serves as a secure way of contacting your server and returning results. In checkout itself, you can offer up information too, with checkout scripts post-purchase.

Note that without integrating a tax service into Shopify itself and using that, none of the tax data or numbers you show off will have any use in the actual order calculations, meaning all your work is simple convenience of showing off an estimation to customers. The tax numbers the customers may actually see will be done by Shopify and/or the integrated tax solution your merchant account uses.