我一字一字地学习汉语 OR 我一个字一个字地学习汉语
Both are correct, but the second one sounds more natural to a native speaker.

个 is omitted only in some idiomatic phrases like 一字一句 or 一字排开(though here 一 doesn't mean the number, but the shape of the number).

In Chinese, it's more idiomatic to use a measure word between the number and the noun in most cases.