How can I clean this older paint roller to make it usable again for painting my walls?
These are consumables

You wouldn't worry about reusing a rubber glove or a paper towel.  If it's pristine, sure... But generally you don't expect to.   

That roller cover is $3, the hot dog is $1 and the brush is $2.  

Stannius makes a good point about buying a quality brush and taking care of it.  That idea actually harkens back to the age when everything was brushed (and oil based) and the best brushes were badger.  It makes even more sense with latex/emulsion.  Won't work with 2-part paints, though.  

It's a waste of your time

You used these supplies for a previous paint job.  At the end of that job, you left them in that state.  That means at that time, you chose to scrap them.

And that's fine. Cleaning them better would've taken more effort.  Your decision back them was probably correct.  Getting them clean is a lot of work and your time also has value.  

Okay then... Here's how you save them

Short-term: you wrap them tightly in plastic, ideally in a ziplock bag and press all the air out.  Air makes them dry and cure.  If they are a 2-part paint such as an epoxy or LPU, you must put them in the fridge or freezer, and you will need a dedicated fridge just for that, because the fumes will contaminate food. I can get 2-4 days tops this way.  This doesn't always work, if your wrapping is poor they will dry in the bag.  If you don't want to chance it, use the following longer term methods. 

Long term: 

Latex/emulsion paints:  you work them in soap and water.  A lot.   You work soap and water into them aggressively, rinse, over and over.  Brush/roll on a clear surface (such as the bottom of the sink) to see how well you are doing.  You are done when they are so clean it is impossible to tell what color you had been painting.   For rollers, some multi-purpose paint scrapers have a hook for squeezing out rollers. 


source

Alkyds, lacquers, and other oil base: Paint the brush out as far as possible on masking paper (don't use newspaper). Splash it with a bit of thinner, work the thinner in, and brush it out again.  Repeat until diminishing returns.  Now put 1/2" of thinner in the bottom of a soup can and work the brush in it.  Dump the thinner, repeat until the thinner runs nearly clear.  Hang the brush by the hole to dry.  Try not to spend more on thinner than the brush is worth, and don't do this on Spare The Air days.  


For rollers, toss them in the trash unless you have a useful application for the paint-contaminated thinner you'll burn through trying to clean them.  For instance I'm fond of painting ultra-thin, and so I save the thinner for later reducing the next coat.  Roll them out on masking paper, put 1/8" of thinner in a clean roller tray, roll through the thinner, squeeze it out as dry as possible, recycle that thinner, repeat until clean.   


2-part coatings of any kind that come in an A and B part: into the trash it goes. Because you definitely will spend 4x the brush's worth in the expensive solvent. 

I just did an LPU project.  I ordered 3 boxes of chip brushes, and 20 of the exotic LPU tolerant rollers.  They are a consumable, like thinner.

How to save dried out plastic roller trays: soak them in water, for a really long time.  I found this out when I abandoned a tray I had used too many times for 2-part epoxy primer.  It filled with rain.  The epoxy coating chipped and blistered, I flexed the tray and the coating just popped out in chips.  This surprised me, as epoxy coatings are used to line the inside of tanks.    That works because roller trays have a very glossy finish paint doesn't want to stick to. The water gave it the nudge it needed to let go.  It would not happen in a properly prepared tank interior.