I'll preface this by saying that I'm not familiar with Rust nor Hearthstone at all, but I do deal with D3D9 and D3D11 on Windows to do similar things. Hopefully this will give you insights how you could approach this. (Closest I've done was code injection on Android)
The most common and robust approach to this is to hook/detour the API functions that the game imports from the renderer backend.
One way you usually do this is by creating a dummy library which overrides/intercepts the system library and passes through every function call to the API except the ones you need. This usually requires you to gather all exported symbols and re-create them, which is a very tedious but rewarding task, as it usually is very stable and can work around things such as DRM.
Usually, since that sits quite low on the application's code stack, it is most efficient for it to be a more general-purpose hook which can load other libraries. Examples would be things like the ASI loader or Reshade on Windows.
Another way would be to do code injection via library side-loading. Essentially, you can simply load a library that performs the code hooks and does necessary renderer API hooking. This is usually done in combination with the previous method (it being a "plugin" loader), however, it is also possible to modify game binaries to call dlopen to load your library and its exported function as an entrypoint (in which case you need to do platform's CPU assembly code for a bit).
Those are the entrypoints for your code. After that, it is all about necessary render backend code that you need to do in order to draw graphics/text/etc.
In C/C++ land I'd just tell you to use Dear ImGui, but seeing as that doesn't exist for Rust, you're kinda on your own.
Same with the API detouring. Ideally, you'd make a plugin loader that does the job for you. Not sure if that exists in Rust yet.
For references, Vulkan overlays such as MangoHUD or ReShade could be useful to help you figure out how to draw stuff on screen.
As for the rest of your code - it can run in a separate thread that does the job for you as the game runs. Or, make a client-server relationship and make the game hook be the server for your info that you need.
No problem. Please do report back!
We really do not have many (if any) Rust alternatives for code hooking and injection and doing stuff with rendering like we do with C++.
Maybe you could finally figure something out we can use for other games as well!