Мой почерк становится аккуратнее, когда я пишу медленнее.

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Breakdown of Мой почерк становится аккуратнее, когда я пишу медленнее.

• мой: my
• писать: to write
• становиться: to become

Questions & Answers about Мой почерк становится аккуратнее, когда я пишу медленнее.

Why is it мой почерк, not моя почерк?

Because почерк is a masculine noun in Russian, so it takes the masculine form of my: мой.

• мой = my (masculine)
• моя = my (feminine)
• моё = my (neuter)
• мои = my (plural)

So:

• мой почерк = my handwriting

What case is почерк in here?

It is in the nominative case because it is the subject of the sentence.

In Мой почерк становится аккуратнее, the thing that is changing is my handwriting, so почерк is the subject:

• Мой почерк = the thing that becomes neater

Why does Russian use становится here?

Становится means becomes / is becoming. Russian uses становиться when something changes into a new state.

So:

• Мой почерк становится аккуратнее = My handwriting becomes neater / is getting neater

This is different from simply describing a state. The sentence is about change, not just a fixed quality.