I am confused sometimes on how to use the verb olmak in the above meaning.

For example:

* He is getting sicker day by day - Günden güne daha hasta oldu / olur / ...
* She is becoming more beautiful every year - Yıldan yıla (her yıl?) daha güzel oldu / olur / ...

Is this a good way of expressing "to become"? And which tense would be the best one?

I think you are referring to linking verbs, particularly to those two. They generally correspond to the turkish suffixes "-er" and "-leş" as in sar-ar-mak and güzel-leş-mek. Thus,

"He is getting sicker day by day" translates Günden güne kötü-leş-iyor..

And "She is becoming more beautiful every year"
Yıldan yıla güzel-leş-iyor.

More examples:

He became mad/insane.
Deli-r-di. / Çıl-d-ır-dı.

The leaves turns yellow.
Yapraklar sar-ar-ıyor.

His face went red.
Yüzü kız-ar-dı.

It is getting more and more difficult.
Giderek zor-laş-ıyor.

Getting complicated. = karmaşık-laş-ma.
Similarly to get complicated = karmaşık-laş-mak
Becoming solid/liquid = katı-laş-ma/sıvı-laş-ma
and to become solid = katı-laş-mak.

Now you can guess the following: çirkin-leş-me(k), keskin-leş-me(k), sert-leş-me(k), gaz-laş-ma(k), aptal-laş-ma(k).



Edit*: The tense remains the same as the relevant one in English. Because the construction is like this:

Adjective + leş + appropriate tense suffix + (personal suffix)
Word root + er + appropriate tense suffix + (personal suffix).

As you see, the relevant tense suffixes stay in their normal position.