How do I say that I went down and came back up the water while drowning?

I would use the word flounder in this case, especially in the sense that it directly references up and down movements.

verb (used without object) 1. to struggle with stumbling or plunging movements (usually followed by about, along, on, through, etc.)

Here's an example:

I watched her flounder in the water for a few minutes before her head finally sunk beneath the surface for good.

Struggle (for breath) is the word most commonly used to describe that kind of situation as defined in Wiktionary:

To strive, or to make efforts, with a twisting, or with contortions of the body. "She struggled to escape from her assailant's grasp".

Keep one's head obove water is rather an idiomatic phrase:

To survive or endure, especially in a situation in which one is struggling to avoid being overwhelmed by adverse financial circumstances

You can easily understand where this idiom comes from.

Fight for one's life (in your question) or fight to catch breath/fight for breath (as suggested by Mari-Lou A in the comment) can be a good candidate, too.

Consider,

gasp for air

: to fight for a breath of air. (After one has been deprived of air.) Walter popped to the surface of the water and gasped for air. The injured dog appeared to be gasping for air

McGraw-Hill Dictionary of American Idioms and Phrasal Verbs

gulp for air

: Fig. to eagerly or desperately try to get air or a breath. Tom gulped for air after trying to hold his breath for three minutes. Mary came up out of the water, gulping for air.

McGraw-Hill Dictionary of American Idioms and Phrasal Verbs

Alternately, battle for air could also fit the bill.

battle

: to struggle tenaciously to achieve or resist something.

OED