Lưng của tôi đau sau khi chạy.

Vietnamese often builds this idea as:

[body part] + [owner] + [state] + [time clause]

So:

• Lưng = back
• của tôi = of me / my
• đau = hurts / painful
• sau khi chạy = after running

A very literal breakdown is:

Back of mine hurts after running.

That sounds odd in English, but it is normal Vietnamese structure.

Lưng means back, especially the back area of the body. In everyday use, it can often suggest the back in general, and depending on context it may sometimes feel closer to the back or the lower back.

If someone wants to be more specific, Vietnamese can use other expressions, but lưng is the normal basic word for back.

Yes, tôi means I / me, but của tôi is the possessive phrase meaning my / mine.

So:

• tôi = I / me
• của tôi = my / mine

In this sentence, lưng của tôi means my back.

The word của is a common possession marker, similar to of in very literal English.

Yes — and in fact, Lưng tôi đau sau khi chạy is very natural and often sounds more casual and conversational than Lưng của tôi đau sau khi chạy.

Both are correct:

• Lưng của tôi đau sau khi chạy.
• Lưng tôi đau sau khi chạy.

The version without của is very common when the relationship is obvious, especially with family members, body parts, and personal belongings.

In Vietnamese, đau behaves like a stative verb or state word. English learners often want to label it as either hurt or painful, but Vietnamese does not always divide things the same way English does.

So Lưng tôi đau naturally means:

• My back hurts
• My back is sore
• My back is painful

Vietnamese does not need an extra verb like is here.

Because Vietnamese often does not use a copula like English to be in sentences describing states.

English says:

• My back is sore
• My back is painful

Vietnamese simply says:

• Lưng tôi đau

So you should not try to insert là here.
Lưng tôi là đau would be wrong.

Sau khi means after in the sense of after doing something.

Breakdown:

• sau = after
• khi = when
• chạy = run / running

Together, sau khi chạy means after running.

It introduces an action that happened earlier. So the sentence means your back hurts after the activity of running.

Usually no, not in this sentence.

To say after running, Vietnamese normally uses:

• sau khi chạy

Without khi, sau chạy sounds incomplete or unnatural in standard usage here.

Other possible ways to express similar ideas include:

• chạy xong, lưng tôi đau = after finishing running, my back hurts
• sau buổi chạy, lưng tôi đau = after the run / after the running session, my back hurts

But for a straightforward learner sentence, sau khi chạy is the best choice.

No. Chạy has several meanings depending on context, such as:

• to run physically
• to operate / run a machine or program
• to rush around
• in some contexts, other extended meanings

But in sau khi chạy, the natural meaning is after running as physical exercise or movement.

It is correct and understandable, but many speakers would more naturally say one of these:

• Lưng tôi đau sau khi chạy.
• Tôi bị đau lưng sau khi chạy.
• Sau khi chạy, lưng tôi đau.

All of these are natural.
Your original sentence is fine; it is just a little fuller because of của tôi.

Both can mean my back hurts, but they are structured differently.

1. Lưng tôi đau

   • literally: my back hurts
   • focuses directly on the body part

2. Tôi bị đau lưng

   • literally: I have back pain / I suffer from back pain
   • focuses more on the person experiencing the problem

Both are common.
If you are talking specifically about the body part itself, Lưng tôi đau is very direct.

By itself, sau khi chạy primarily means after running, so it clearly gives the time relationship.

In real conversation, people will often also understand a likely connection — that the back pain came after, and perhaps because of, the running. But grammatically, it does not state the cause as explicitly as because of running would.

If you want to make the cause clearer, you could say something like:

• Lưng tôi đau vì chạy. = My back hurts because of running.
• Sau khi chạy, lưng tôi bị đau. = After running, my back started to hurt.

A careful pronunciation guide is roughly:

Lưng của tôi đau sau khi chạy.
→ lɨng / kuə / toi / ɗaw / saw / xi / cạy
(This is only approximate.)

A few useful pronunciation notes:

• lưng has the ưng sound, which is unfamiliar to many English speakers.
• của is often pronounced more like của / củaa, and in fast speech it may sound reduced.
• đau rhymes roughly with ow in cow, but starts with đ, a plain voiced d sound.
• khi begins with the kh sound, like a breathy h/k sound, not like English k alone.
• chạy has a heavy falling glottalized tone in the standard southern spelling-based description; tone realization differs by region.

If you want, I can also break the whole sentence down tone by tone in a learner-friendly way.

Yes. Vietnamese often allows time phrases at the beginning.

So these are both correct:

• Lưng tôi đau sau khi chạy.
• Sau khi chạy, lưng tôi đau.

The second version puts more focus on the time setting: after running.
This is very natural in speech and writing.