Learn Hardcore Japanese: watasi no yatin ha amari takaku arimasen.

家賃 is read やちん (yachin). It’s a compound of 家 (house) and 賃 (rent), so it means rent (specifically, the rent you pay for a house or apartment).

What is the function of the particle は in 家賃は?

は is the topic marker. It tells the listener that 家賃 (rent) is the topic of the sentence—“As for the rent, …”. It doesn’t mark the subject (which is often the same), but highlights what you’re talking about.

What does あまり do in this sentence?

When used with a negative verb or adjective, あまり means “not very” or “not much.” Here, it weakens 高くありません, so あまり高くありません means not very expensive (literally, “the rent is not very high”).

How is the adjective 高い negated in 高くありません?

高い is an い-adjective. To form the polite negative:

Drop the final い → 高 (taka)
Add く → 高く (takaku)
Attach the polite negative ありません → 高くありません (is not high/expensive)

Why does the sentence use ありません instead of ない or ないです?

ない is the plain (casual) negative form.
ないです is colloquial and somewhat mixed register.
ありません is the fully polite negative of ある and keeps the sentence in polite/formal speech.

Can you omit 私の and still be natural?

Yes. If context makes it clear whose rent you’re talking about, you can simply say:
家賃はあまり高くありません。
Japanese often drops pronouns when they’re understood from context.

How would you say “my rent is very expensive” instead of “not very expensive”?

Use a positive degree adverb like とても or すごく, and the polite copula です:
私の家賃はとても高いです。 (My rent is very expensive.)