I have an article written with keywords but I don't know if I should backlink the keyword to my website or if I should just backlink the text showing my website name.
you can link them however you want, 1 or 2 links will not be a problem...
But when you have many links pointing to the same URL you need to be mindful of the anchors, especially the exact match ones as those are the most sensitive and can trigger google penalties the fastest.
And can I backlink my home page and then some other different pages, is that good?
you can link to more than 1 page, but the common practice is to use 1-3 links per article, more than this you will be diluting the power that each page gets, and I don't think you want that
When I was in this business I would almost always (like 80-90% of the time) link to 2 different pages (to the homepage with branded anchor text, and to an inner page with a generic anchor text), but that was several years ago, I'm not sure what practices are working well nowadays...
But yeah, you can definitely link to more than 1 page, just don't overdo it because each one of the pages that you link to will get less and less link juice (authority, page rank, whatever you want to call it, it's basically the same thing)...