Last Sunday, I enjoyed playing Duolingo story lessons during my live stream on Twitch.

Duolingo is a kind of easy for me, so I play this app for entertainment purposes.

In the Duolingo lessons, some funamusingly named characters appear.

Last Sunday, I enjoyed doing some Duolingo story lessons during my live stream on Twitch.

Duolingo is a kind of easy for me, so I playuse this app for entertainment purposes.

"for entertainment purposes" sounds a bit stiff, more casual would be "just for fun".

In the Duolingo lessons, some funinterestingly named characters appear.

I can see someone saying 'fun named' in speech, but it looks odd in writing.

Basically, they know each other.

When a sentence starts with 'basically' it sounds like you're explaining something (in this case, like you're explaining why they have interesting names). This sentence feels like it is missing something.

Last Sunday, I enjoyed doing some Duolingo story lessons during my live stream on Twitch.

When I hear "I enjoyed Duolingo story lessons", I feel that I don't know `how` you enjoyed it. I assume you enjoyed the lessons by `doing them`, thus the correction.

For example:
X. I enjoyed a movie last night.
sounds a little weird.

0. I enjoyed watching a movie last night.
sounds natural.

Duolingo is a kind of easy for me, so I play this appwith it for entertainment purposes.

It's not quite natural to 'play an app', but it is natural to 'play with an app'.

In the Duolingo lessons, some fun named characters appear.

Did you mean to say the names are funny? This is one way you could say that.
* In the Duolingo lessons, some [funnily] named characters appear.

Basically, they know each other.

How do they "know each other", If you want to maintain suspense:
* Basically, they know each other in ways that you'd never imagine.

I will write about somthese memorable characters in future posts.

Also:
I will write about some [of these] memorable characters in future posts.