When installing a Partial.ly checkout button onto your Shopify store, Partial.ly will offer several button images to choose from.

If you'd like to use your own image, just select the 'Use a custom image for the checkout button' checkbox on the integration tool and add the image URL for the image file that you'd like to add.
checkout-button-custom-image

It is recommended to use the Shopify files section of your Shopify admin to add the image. Upload the file and then copy and paste the resulting code in the image URL field on the integration tool for the offer you want to add to your site.