Loading is the process the application performs when it opens a file. Depending on the application and the file, aspects of the loading process could include:
Rendering appropriate text in bold, or italics etc for a word processing app with a document file.
Performing formulae calculations and rendering charts for a spreasheet program with a csv file.
Displaying populated fields in a form-base app that uses a database
Uploading is the process of transferring a file (or data) from one location to another. Generally, when the file is being moved from a local to a remote store, it is called uploading, and when the file is being transferred from the remote location to your own store it is called downloading.
You would generally use load to describe this situation you've mentioned. The significant aspect of your process is seeing the file properly rendered in the browser after the browser has identified the H1, div, bold etc html codes. If the file is not on your machine then technically, it is downloaded before it is loaded, but there's no need to explicitly mention this.
If your program just copied the file from the local computer to a remote machine (without opening it), then you would use upload.