If you really want to strain, straining with cheesecloth leaves as much of the grease behind as possible while picking up the bacon particles. Paper towels over your jar/receptacle of choice will work, but will soak up a good amount of the grease.
I usually bake my bacon and pour off the grease to cook with. I think Mark has an article about how long bacon grease is safe for, but honestly, I keep it for months in the fridge sometimes. I always had more than enough when I was cooking bacon weekly. I don't have it as often now (too costly).
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Just pour it out into a ramekin and do your eggs in the greased pan that remains. As you collect more and more fat in your ramekin you can use that to fry things off in or "wet up" veggies after steaming.
Don't bother straining - just eat it all. If you were bothered, you can pour into a ramekin, fridge 'til solid, scrape off the clean top and discard the settled muck underneath.