Interesting that two of your three respondents so far would never use "keep you informed". I certainly do use it, but as Michael says it's probably the most formal of the options here - so in fact I probably normally write "I will keep you informed" rather than using the contraction "I'll".

Hello! I agree with Skye, but I would like to add the following; I would use informed as a verb in the simple past tense "I informed the family that they had won the competition" or as an adjective "It is important to look at all the facts when making an informed decision".

I would say I’ll keep you posted, or if more formal, I’ll keep you updated. I would not use “I’ll keep you informed”.

Either is fine. "Informed" is a little more formal, a little less modern than "posted".

Many thanks, all the answers were very helpful!