I believe the default (not on my gaming PC so cannot verify the exact location) for BnS is to save pictures in C:\Users\<Windows User Account Name>\Pictures\BNS\Screenshots. This should be the same for all computers running Windows 7 or later. Earlier OS will be different and I dont remember where the User Account pictures folder is. For Windows 7 and later you can go to the Picture folder directly in Windows Explorer without having to navigate the Users folder structure.
I have been using tinypic.com forever but it changed awhile back where you have to watch a simple ad to get a security code to upload. After a little searching I found http://postimage.org/. Free with no ads and no security code so I am going to outline how to use that.
Go to http://postimage.org/ to upload your image. It is straight forward so no need to detail that part.
Once you have uploaded your picture you will want to use the Direct Link or one of the Hotlink for Forums option.
For Direct Link, click Copy to Clipboard button right next to the link. In the forum post editor click Img button then Ctl + V to paste the link. All done

For Hotlink you can just paste the link directly into the forum post editor. Both Hotlink 1 and 2 wraps the image link in a URL tag that will lead back to postimage.org. Hotlink 2 goes a step further and adds image hoster text to the URL.
Which ioption is better? I will be using direct link as I do not like the image becoming a URL but that is just me.