Car starting Issue

Several weeks ago, I noticed that my car would occasionally struggle to start. The car was gradually getting harder to start. The battery seemed like it was dying. The car would grind slowly as it was starting, but then eventually start. Finally one time, it would not start. Then I tried again, and it barely started, and I drove straight to an auto store and bought a new battery.

Since then, it's been mostly OK, but a week or so ago, it was the same thing all over again. Grinding sound like the battery was dying. Now, this morning, it would not start on the first 2 tries - not enough power to get it started (like the battery was weak). I tried again and it barely started.

What is going on here? I assume the battery I had was bad. The guy at the auto store said it was an original battery (for a 2005 car?!). I don't remember ever replacing it, but its been a long time and I could have forgotten. But could this new battery also be bad or is there another problem like maybe the alternator not charging the battery?