Others have suggested quite a few circuits, so I won't.
When I was into restoring old radios, I used to broadcast OTR (Old Time Radio) programs around the house (I have quite the collection of shows). I wanted something with good audio quality, that didn't drift in frequency, as many simple transmitters do. I bought the SSTRAN AM transmitter kit, which has excellent audio, and uses a PLL. Sadly, you can't buy the kit any more. If you look around, you may find a second hand, assembled unit.
I had a spare PC running a free radio automation program called ZaraRadio. I created playlists, so that certain radio programs would play at a particular time, like a real radio station. Of course, I could always queue up whatever show that i wanted to listen to.
I used Virtual Audio Cable to route the audio through some processing software on the PC (a little compression, a little EQ), then out to the transmitter. It sounded great.
It was a fun project, and it was fun to listen to authentic old radio shows on an old radio with glowing tubes
Now I just load up my phone with the MP3 files and listen.
FYI for those in the US - transmitting on the AM/MW band is covered by FCC Part 15.219. The two most important points are: (a) The total input power to the final radio frequency stage (exclusive of filament or heater power) shall not exceed 100 milliwatts. (b) The total length of the transmission line, antenna and ground lead (if used) shall not exceed 3 meters.
Speaking of OTR, if anyone is interested, the best and most complete collections of shows is put out by the Old Time Radio Researchers (OTRR). When I was into collecting shows, we mailed CD's to each other. Now, you can find it on the Internet Archive. You will find other shows, posted by others, but they usually are a mixed bag. The stuff put out by OTRR is top notch.
Footnote to all of this - Years ago, when I had all of the above running, I was contacted by an FCC field agent. It seems that a local station turned me in. After a lengthy phone conversation, and some emailed documentation, the agent told me that I was compliant with the rules, and that he would close the case. He also complemented me on my setup.