Ég hef verið að bæta ferilskrána mína, af því að ég vil sækja um betra starf.

This is a very common Icelandic way to express something close to English have been improving.

It is built like this:

• ég hef = I have
• verið = been
• að bæta = improving / to improve, as part of the construction

So Ég hef verið að bæta ferilskrána mína means the action has been going on for some time, or recently, up to now.

A useful comparison:

• Ég bæti ferilskrána mína = I improve my CV / I am improving my CV, depending on context
• Ég er að bæta ferilskrána mína = I am improving my CV
• Ég hef verið að bæta ferilskrána mína = I have been improving my CV

After vera in this progressive-type construction, Icelandic uses að + infinitive.

So:

• vera að bæta = to be improving
• hafa verið að bæta = to have been improving

Here, að is part of the pattern. It is not exactly the same as English to, even though it often looks similar in dictionary forms.

Here bæta means improve or make better.

That verb can have slightly different meanings depending on context. For example, it can sometimes mean add or make up for, but with ferilskrá it naturally means improve / update / polish the résumé or CV.

So in this sentence, the idea is not just adding something randomly, but making the CV better.

Ferilskrána mína is the normal Icelandic way to say my CV in this kind of sentence.

A few things are happening:

• ferilskrá is a feminine noun
• ferilskrána is the definite accusative singular form = the CV
• mína is the possessive my, agreeing with the noun in gender, number, and case

So literally it looks a bit like the CV my, but in natural English it is simply my CV.

Icelandic very often uses:

• noun + definite article + possessive

So:

• ferilskráin mín = my CV
• ferilskrána mína = my CV, as the direct object

Mín ferilskrá is possible, but it sounds more emphatic or contrastive, like my CV as opposed to someone else’s.

Because it has to match ferilskrána.

The noun is:

• feminine
• singular
• accusative

So the possessive pronoun must also be:

• feminine
• singular
• accusative

That gives mína.

This kind of agreement is very important in Icelandic. Possessives behave a lot like adjectives in that respect.

Af því að means because.

Yes, it is very common and natural. It is one of the standard ways to introduce a reason.

Other ways to say because include:

• því að
• vegna þess að

But af því að is very normal and everyday.

Because af því að introduces a subordinate clause.

In main clauses, Icelandic often follows the verb-second pattern. But in subordinate clauses, the word order is usually more straightforward:

• conjunction
• subject
• finite verb

So:

• af því að ég vil... = because I want...

not:

• af því að vil ég...

That second version sounds wrong in standard Icelandic.

Because vilja is a modal-type verb, and modal verbs usually take a bare infinitive in Icelandic.

So you say:

• ég vil sækja um = I want to apply
• ég get komið = I can come
• ég skal hjálpa = I will/shall help

not:

• ég vil að sækja um

So even though English uses to after want, Icelandic does not use að here.

Sækja um is a fixed verb expression meaning apply for.

So:

• sækja um starf = apply for a job
• sækja um skóla = apply to a school
• sækja um leyfi = apply for permission

The little word um belongs with sækja here. You should learn it as one unit:

• að sækja um

If you learn only sækja by itself, you may miss this meaning, because sækja alone can have other senses in other contexts.

Because starf is a neuter singular noun, and the adjective must match it.

The comparative of góður is betri, but it changes form depending on gender and number. With a neuter singular noun like starf, the correct form is betra.

So:

• betra starf = a better job

This is also the object of sækja um, but since neuter singular often looks the same in nominative and accusative, the form still comes out as betra starf.

Yes, you could, but the nuance is different.

• Ég hef verið að bæta ferilskrána mína = I have been improving my CV

  • focuses on the process
  • suggests ongoing or recent activity

• Ég hef bætt ferilskrána mína = I have improved my CV

  • focuses more on the result
  • can sound more completed

So the original sentence is a better match if you want to emphasize that this has been something you have been working on.