How to achieve this particular photo filter effect?
I would like to learn how to achieve this effect where the only color used to represent an image is blue (or any color, for that matter). It seems useful for printing processes where only one color can be used (i.e. Risograph or screenprinting). Thanks for your time.
If you are really interested in the printing more than what you can see on screen....
Your samples are merely black and white photos printed with a colored ink. Nothing more complex. For print production, you do not always need to actually have a colored image on a monitor. You can simply use black and white and tell the print provider to print it in color.
This is absolutely the case for the bottom sample. Your bottom sample is simply a 1 color print job. That one color being blue. The artwork is set up back and white and blue ink is used on the press. That's all there is to it.
The top sample most likely is still using black and white photos. You can set the color of a black and white photo in many apps such as Adobe InDesign, Adobe Illustrator, QuarkXPress. You simply place the photo then pick the color to substitute for black.
If you want to see the colored image on screen, there are half a dozen ways to do that depending upon the software you are using. You fail to mention any software so, I won't guess at telling you how not knowing what software you have available, but ultimately it all comes down to a standard black and white image.
Note that setting the images up as colored in an app such as Photoshop has to be done a specific way if you want the print production to be correct. You can't simply create a blue image and expect it to print as a one color print job. Using gradient maps and color overlay layers still retains the full 4-color processes output (which is fine if you want a 4-color print job).