Cover letter and feelings - use or avoid A couple of cover letter use: 


I believe that I could ..
I am confident 
I am very interested 


What do you think about them? Should they be mentioned with evidences or avoid since they are feelings and may indicate uncertainty?
Personally, I think your sentence fragments all suggest both a lack of evidence and a lack of confidence.  But more importantly, I think you're simply asking the wrong question.

Instead of focusing on the detailed language of your cover letter, focus on the content.  Who are you?  What have you done?  What makes you an asset to your target departments?  Instead of expressing your interests, demonstrate your accomplishments and your vision.  Instead of stating your beliefs or your confidence, show the reader clear evidence of your expertise and your impact.  Show, don't tell.

The advice that I give my own students and colleagues when they apply for graduate school or for jobs or for tenure is to aim for the next target.


Graduate applications should not say "Please admit me" but rather "I will become an independent researcher.  Let me get on with it."
Thesis proposals should not say "Please let me pass" but rather "I will have a strong PhD thesis. Let me get on with it."
Thesis defenses should not say "Please let me graduate" but rather  "I will get an academic job. Let me get on with it."
Job talks should not say "Please hire me" but rather "I will get tenure. Let me get on with it."
Tenure packages should not say "Please give me tenure" but rather "I will be a full professor. Let me get on with it."


And the way each application should "say" its message is not by stating your interest or belief or intention or confidence that you'll pass the next stage, but by presenting clear and compelling evidence that you'll pass the next stage.  And of course you never want to actually suggest the impatience implicit in the phrase "Let me get on with it"; rather, you want to convince your audience to let you get on with it.

Don't try to convince that reader that you believe that you'll be successful.  Make the reader believe that you'll be successful.  Show the reader that you will be successful.