Suppose someone really truly wants to believe in Hashem, but is simply unable to. Even after much trying and learning Torah, he still is simply unable to believe in God. Would that person be considered a kofer? Is there anything that he can do to not be a kofer?
"God exempts a person under duress" (Bava Kama 28b).
It is certainly required to believe in Hashem, but if you try hard to believe and cannot believe, then I would say you are under duress and, for now at least, are exempt from the mitzvah to believe in Him. For the same reason, I wouldn't worry about whether you have the status of a kofer.
However, that does not mean you should stop trying. Have you read books meant to convince you that Judaism is true? Beyond a Reasonable Doubt by R' Shmuel Waldman? Soul Searching by R' Yaakov Astor? Other books, such as the stirring classic R' S.R. Hirsch's Horeb, or contemporary spiritually-oriented works such as Bilvavi Mishkan Evneh (which can be read online in English for free) or the books of R' Shalom Arush or R' Lazer Brody? The teachings of Rebbe Nachman of Breslov? The writings of the Lubavitcher Rebbe and his followers? I mention quite a few, because perhaps if you study everything, you will find a particular author who arouses your heart and inspires you to believe.
Have you spent a significant amount of time praying in your own words each day, asking Hashem to help you believe? There's nothing wrong with asking Him for a sign that He exists. You may be surprised by the result. "Everything good can be attained through prayer: Torah, devotion, holiness... everything good in all the worlds!" Likutey Moharan II, 111.
Have you consulted rabbis known for giving spiritual guidance? Nowadays, many well-known rabbis will respond to emails, even if you're not located near any rabbis.
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"emunah" is the term you are looking for. "Emunah" is the basic understand of a jew that everything written in our sources is true an g'd is constantly with us. "emunah" must be learned specifically (like halacha), it cannot be attained otherwise. Learning chumash with rashi or a masechet in talmud might be intellectual interesting but first your emunah must be strengthened. "From where do we know that judaism is right", that everything written is true? A famous answer is the "iggeret teiman" (letter to the jemenite jews) written by Maimonides addressing specifically this question raised by the jemenite jews that wanted to convert to islam. A modern approach which had enormous impact on my believe (emunah) was "Letters to a buddhist jew" by Rav Akiva Tatz. This book answers specific questions raised by a atheist buddhist jew who even did not want to believe - he had to give it a try because of his wife! ;-) He is philosophically experienced and so is the level of this book, but it is fitting for all levels. And there is a whole chapter about "emunah". A further very good source a shiurim of "machon meir" which are specifically addressing this topic: http://www.english.machonmeir.net/torah-lessons-archive?view=jmultimedia&catid=18
Or the most famous book of all times when it comes to emunah: the kuzari. No need to comment. Every jew should read that book. (the vilna gaon had always two books with him: "Messilat jesharim" and "the kuzari". And he read them constantly).
But I fear your question was a technical halacha question and I simply missed the point totally.... ;-) sorry, if that is the case.