While you can upload videos onto Flickr in bulk from the browser, a better option is their desktop based Flickr Uploadr software - recommended when you have multiple videos to upload or the size of individual videos is very large.
There are some extra advantages. With Flickr Uploadr, you can attach tags, titles and descriptions to all your videos, as well as set privacy options. And the software works in parallel meaning you can prepare a second set of videos for uploading while the previous one is still getting uploaded onto Flickr.
Flickr Uploadr software, currently available on the Flickr tools page, doesn’t include support for videos but there’s a new official version (v3.1) that allows uploading of photos as well as videos. Here are the download links.
FlickrUploadr-3.1.0-en.exe [Windows] FlickrUploadr-3.1-en.dmg [Mac]