Learn Russian: Он забыл указать твои достижения в отчёте. - He forgot to indicate your achievements in the report.

The verb забыть (to forget) indicates a failure to perform a specific action. In Russian, verbs expressing forgetting, succeeding, or managing to do something are almost always followed by a perfective infinitive. This is because they focus on the result (or lack of result) of a specific, single action.

What case is твои достижения and why doesn't its ending look different?

It is in the accusative plural case because it is the direct object of the verb указать (to indicate). The dictionary form is the neuter noun достижение (achievement). Because inanimate neuter nouns don't change form between the nominative and accusative, the accusative plural (достижения) looks exactly the same as the nominative plural.

Why is the prepositional case used for в отчёте instead of the accusative?

We use the prepositional case (в + отчёте) because it indicates a static location, answering the question where the achievements belong. The dictionary form is отчёт (report), and adding the -е ending shows that the information was supposed to be located inside it.

If he repeatedly forgot to do this, would both verbs become imperfective?

No. If this was a habit, you would change the first verb to the imperfective забывал (he used to forget). However, the second verb указать would remain perfective. This is because every individual time he forgot, he was still failing to complete the specific result of indicating the achievements.