Мой почерк становится аккуратнее, когда я пишу медленнее.

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

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

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.

Because the verb is становиться, which is the normal verb meaning to become.

The -ся here is not something you translate separately as self. It is just part of the verb.

Forms:

• я становлюсь
• ты становишься
• он / она / оно становится
• мы становимся

So становится is simply the 3rd person singular form, agreeing with почерк.

Because аккуратнее is the comparative form: neater / more neat.

The sentence is comparing two situations:

• when I write at one speed
• when I write more slowly

As a result, the handwriting becomes neater, so Russian uses the comparative:

• аккуратный = neat
• аккуратнее = neater

After становиться, Russian often uses the comparative in this kind of sentence:

• становится лучше = becomes better
• становится проще = becomes simpler
• становится аккуратнее = becomes neater

In this sentence, it functions like a comparative predicate word after становится.

For learners, the easiest way to think of it is:

• аккуратный = neat
• аккуратнее = neater

Russian comparative forms like аккуратнее are often used after verbs such as стать / становиться without changing for gender, number, or case.

So even though почерк is masculine, you still say:

• почерк становится аккуратнее not
• почерк становится аккуратнейший or something gender-marked

Here it means more slowly.

That is because it goes with the verb пишу:

• я пишу медленнее = I write more slowly

Russian uses the same comparative form for:

• the adjective: slow/slower
• the adverb: slowly/more slowly

So медленнее can mean either slower or more slowly, depending on context.

Here the context is a verb of action (пишу), so it is understood as an adverb:

• I write more slowly

Because когда means when, and it fits a situation that is presented as something that regularly happens.

The sentence means something like:

• Whenever I write more slowly, my handwriting becomes neater.

If you used если, that would sound more like a condition:

• If I write more slowly, my handwriting becomes neater.

Sometimes both are possible, but когда is very natural when talking about a repeated or typical relationship between two actions.

Because когда я пишу медленнее is a subordinate clause.

Russian normally separates subordinate clauses with a comma:

• Мой почерк становится аккуратнее, когда я пишу медленнее.

This is similar to English, where a clause with when may also be separated depending on position and style.

Because пишу is the imperfective present, which is used for:

• habitual actions
• repeated actions
• ongoing processes
• general truths

Here the sentence describes a general pattern:

• When I write more slowly, my handwriting becomes neater.

So the imperfective is the natural choice.

Compare:

• я пишу = I write / I am writing
• я написал = I wrote / I have written
• я напишу = I will write / I will finish writing

The sentence is not about one completed piece of writing, but about a regular tendency.

Yes. Russian word order is flexible.

You can also say:

• Когда я пишу медленнее, мой почерк становится аккуратнее.

This means the same thing.

The difference is mainly in focus:

• Мой почерк становится аккуратнее, когда я пишу медленнее.
  Starts with the main idea: my handwriting becomes neater
• Когда я пишу медленнее, мой почерк становится аккуратнее.
  Starts with the condition/circumstance: when I write more slowly

Both are correct.

Почерк specifically means handwriting in the sense of a person’s writing style or penmanship.

So it is not the same as every meaning of English writing.

Examples:

• красивый почерк = beautiful handwriting
• неразборчивый почерк = illegible handwriting

In this sentence, почерк refers to how neatly the letters are formed.

Yes, there are other natural ways to express a similar idea, for example:

• Если я пишу медленно, почерк у меня аккуратнее.
• Когда я пишу медленно, у меня почерк аккуратнее.

But the original sentence is good, natural Russian and slightly more polished because it clearly expresses becoming neater:

• становится аккуратнее

They are both comparative forms made from adjectives/adverbs.

• аккуратный → аккуратнее = neat → neater
• медленный / медленно → медленнее = slow / slowly → slower / more slowly

A useful pattern is:

• X становится + comparative
  • становится лучше
  • становится труднее
  • становится аккуратнее

and

• verb + comparative adverb
  • говорю громче = I speak louder
  • иду быстрее = I go faster
  • пишу медленнее = I write more slowly

This sentence nicely shows both patterns at once.