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I want to see the total weight of the order on my order page (Shopify Classic Admin)
You can submit the cart's total weight as a cart attribute to your shop's back-end. If you do that, the total weight will be displayed on the order details page in your shop admin as an Order Note:

How do you do this
You will need to edit the template cart.liquid in your theme.
To edit your cart.liquid template, go to Theme → Template Editor:

On the Template Editor page, look under Templates for cart.liquid. (The .liquid extension is omitted in that view.)
Click on 'cart' to open cart.liquid in the online code editor.
Find this code:
<form action="/cart" method="post" id="cartform">
(That form tag may look a bit different in your theme, it may not have an id attribute.)
On the next line, add this:
<input type="hidden" name="attributes[weight]" value="{{ cart.total_weight | weight_with_unit }}" />
(You know HTML a bit? What we are doing here is adding a hidden form field inside the cart form.)
Save.
How do I see the total weight of the order on my order page? (Shopify 2 Admin)
You can submit the cart's total weight as a "cart attribute" to your shop's back-end. If you do that, the total weight will be displayed on the order details page in your shop admin as an Order Note. It will look like this:

To do this, you will need to edit the template cart.liquid in your theme in a few easy steps:
1. To edit your cart.liquid template, go to the Theme section of your admin panel.
2. Click the Template Editor
button.
3. On the Template Editor page, look under the Templates folder for the cart.liquid file.
4. Click on cart.liquid to open it in the online code editor.
5. Find this code (you can hit CTRL-F or COMMAND-F and search for the code):
<form action="/cart" method="post" id="cartform">
Note: this form tag may look a bit different in your theme, as it may not have an "id attribute."
6. On the line directly after the code above, add this:
<input type="hidden" name="attributes[weight]" value="{{ cart.total_weight | weight_with_unit }}" />
7. Click Save.
That's all there is to it!
Now the total weight of an order will be displayed on the Order details page in your shop admin as an Order Note.
