How to upload photos/files to iCloud like with Dropbox's application? I've recently extended my iCloud storage to 20GB and now I am wondering how to upload files to it from my iPhone.

Is there any way to do it as it is in dropbox's application? Just to open the folder that you want and upload the files.

If not, are there any alternatives?
I'm sorry, you can't manually upload files to iCloud from your iPhone. Since iOS 8 Apple introduced iCloud Drive, which makes it possible to upload any file with a Mac (on Yosemite). 

You can only access those files on an iPhone when you're, for example, in the Dropbox app.