"I don't think Chieko likes Mayumi."
Your TL of that is:
Your translation is good if the purpose of the lesson is to learn to use 「思う」 in the affirmative form even though the sentence in question is clearly in the negative form "I don't think".
As a Japanese sentence, it is impeccable. Its meaning is not ambiguous at all unless you say it in a situiation where talking about who Chieko likes or dislikes is clearly irrelevant. Japanese is an incredibly contexual language; Take that from a veteran native speaker.
If you wanted to use 「私は」, I would suggest you place it right in front of the verb at the end instead of at the beginning of the sentence. That would be:
It is, of course, grammatical to say:
Mentioning all three persons at the beginning makes it look/sound kind of crammed, which is why I used a comma after the 「私は」.
「私は」 is not required at all in Japanese though it might take time for you to get used to it.
Could it instead mean that Chieko thinks Mayumi doesn't like (something)?
Again, it could if it is said in a situation/context where the topic of the discussion is who likes or dislikes something (instead of who likes or dislikes someone), which is why I have to type the same sentence here everyday: "Japanese is a contextual language".
The exact same phrase/sentence can mean completely different things.
Finally, your sentence:
means a different thing than:
To say the former sentence, you need to have been talking about multiple people who might or might not like Mayumi. Then you are saying that out of all those people, it is Chieko who does not like Mayumi.
To say the latter, no particular context is needed. You are just saying that Chieko does not like Mayumi.
If nothing I am saying makes sense to you, do not worry too much about it now. You will get used to our particles in a few years. As if I had met someone who had mastered the particles in a few years...