Can someone tell me how to use this phrase and how I should translate it? Some examples would be helpful. Thanks!
"この度" (このたび) can mean several things, including "this time", "soon", or "next time". It is fundamentally very similar to the word 今度（こんど）, though この度 seems to more commonly used in some places like formal greetings and news. I've never heard or used this word in informal conversation (For a comparison of these two words and some other similar words, see this page here, in Japanese)
Here is one common phrase which uses この度:
Congratulations on your wedding
Here この度 means something like "today" or "this time", but in a natural translation to English it doesn't have to translate to any specific word. (You could probably say "Congratulations on your wedding today", though).
So "この度の + Noun" is used when you want to modify a noun to be associated with one of the above meanings I mentioned ("this time", "soon", "next time", etc). Most commonly, I think it is used when referring to a specific instance of something, which is the "this time" meaning, though this can be translated just as "this ~", or omitted completely.
To give an example, let's say someone who worked at a store accidentally left the door open and during the night a great deal of money was stolen. When the manager came to him, he could say something like:
I'm extremely sorry for what happened in this situation <= More literal, less natural
I'm extremely sorry for what happened <= More natural
Here, we can translate この度の事 as "this situation" (although "事" is technically closer to intangible "thing"), but this English sentence sounds a bit awkward to me. Removing that gives a more natural translation.
Another meaning of この度の could be when that phrase itself is being referred to:
I don't know what "この度" means.