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?
I'm going to assume that this 'inability to believe in God' comes from a conviction that God doesn't exist. The question is, should a person be faulted for disbelieving, if he thinks that believing in God is philosophically unjustified?
First off, I should mention the Rashash to Shabbos 31a, who writes that a person is only considered a heretic after fully analyzing every side of the issue, which I doubt most people have. Until then, they are merely 'non-believers', which isn't such a problem. I'll quote his words in case I'm misinterpreting them:
דכופר לא מיקרי אלא אחר החקירה בכל חלקי הסותר, אבל זה לא חקר ולא נוכח אלא שלא שלא היה מאמין
However, I'm not sure if this should be considered agreed upon by most poskim, as it seems to contradict R. Elchonon Wasserman in the first piece in Kovetz Maamarim as well as R. M.M. Shach to Hil Teshuvah 5:5 - see below
There are two consequences of being a disbeliever/heretic: (1) according to the Rambam (intro to Cheilek and Hilchos Teshuva ch. 4), at least, a heretic had no share in the World to Come, no matter how good of a person he may have been otherwise (2) regarding many (but not all) halachos, a heretic is treated as a non-Jew and other Jews are obligated to hate him and possibly even throw him into a pit to let him die (ibid).
Regarding (2), most contemporary poskim, most notably the Chazon Ish, have assumed that at least some, if not all, of those halakhos are no longer applicable today to a person who doesn't believe in God, because we'd attribute it to his upbringing or the inability to be properly philosophically convinced due to uniquely contemporary circumstances. See Chazon Ish Y.D. 1:6, 2:16, 2:18, as well as in 2:28 where he writes that this was also the opinion of the Chofetz Chaim. The Chazon Ish is actually not the first to pasken like this - I believe that it is the opinion of R. Yaakov Ettlinger in Shut Binyan Tzion Hachadashos 23 (though I'm not sure what to make of that title). See also R. Moshe Feinstein in Iggros Moshe O.C. 4:91:6, as well as many other later poskim who have accepted this opinion, though there will always be dissenters here and there.
There is an even earlier body of literature regarding (1). The Raavad to Hilchos Teshuva 3:7 assumes that if someone believes in heresy merely due to an honest mistake, he should not be considered a heretic. This position appears to be agreed upon by the Radvaz 4:187, as well as R. Yosef Albo in Sefer HaIkkarim 1:3 (though he writes that such a person still sinned).
However, there is considerable debate as to the position of the Rambam himself on this matter and whether the Raavad is correct. Abarbanel (Rosh Amana ch. 12) writes that one who doesn't believe in God would certainly not have access to Olam Haba, even if it's through no fault of his own, just as someone who drinks poison thinking it is wine would still die from the poison. There's an oft-quoted opinion of R. Chaim of Brisk that one who falls to apikores 'by accident' is still an apikores. While this isn't in R. Chaim's own writings, it's quoted by his son R. Velvel, by R. Moshe Shmeul Shapiro, by R. Yichiel Michel Feinstein, and comes up in other random places, though it's meaning and veracity are still being debated. R. Elchonon Wasserman (Kovetz Maamarim pg. 19) interprets this statement very differently: there's no such thing as being a heretic 'by accident', since the truth of God is so obvious that only a wicked person would deny it. (I personally have a lot of trouble accepting this though, especially regarding the other 12 principles besides belief in God).
"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.
"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.
Try different avenues. Torah study about an ox goring a cow is not going to answer your question of faith. There's even a famous disagreement between the Rambam and the Ramban whether belief in Gd precludes the commandments of the torah altogether.
Another avenue well worth pursuing is to study the divine wisdom exhibited in nature (as brought in the chovos halevavos shaar bechina). This is becoming increasingly revealed by modern technology and is building an enormously powerful argument to design in my opinion.
(I prepared a brief "tour" on this subject here for whatever it's worth)