The reason historians try to avoid using terms like "homosexual" or "bisexual" is because they carry meanings in the present that don't make sense to apply to the past. The most famous work on this is Bray, Homosexuality in Renaissance England. Homosexual was a word that entered the English lexicon in the 1890s. Words that we would now associate with male homosexuality like sodomite and bugger conveyed broader meanings in their early modern context than we would want. Sodomy was very much a sin or vice that accompanied or derived from other sins. Someone described as a bugger did not mean simply that he was a man who anal intercourse with other men (or animals), at least in the context of literary descriptions. Instead, the author was likely trying to covey that the subject committed multiple sins, including adultery, pride, gluttony, or drunkenness. A criminal indictment might be more precise, as Coke wrote in his institutes that there needed to be penetration between man and man, man and beast, or woman with beast (Institutes of the Lawes of England, Volume 3, pp. 58-59, 1797 ed.). Despite the technical definition in his Institutes, Coke better describes the popular sense of the word when he tried to explain the cause of sodomy. Coke wrote that "the sodomites came to this abomination by four means, viz. by pride, excesse of diet, idlenesse, and contempt of the poor." There is much going on then in political or poetic works that deal with terms close to homosexual or homosexuality. None of them are precise enough for most historians to feel comfortable referring to people in the past with modern sexual identities.

When we discuss homosexuality in the modern sense, we are not discussing it as a production of sin, but instead as description of our identity. The sense of personal identity that accompanies our modern LGBTQ language would not make sense in an early modern time. The closest approximation we can come to in the case of King James is that he was likely sexually attracted to both men and women, though I would hesitate to apply the label "bisexual".

The classical opinion of King James and his sexuality is summed up in Laurence Stone's Causes of the English Revolution. He writes:

As a hated Scot, James was suspect to the English from the beginning, and his ungainly presence, mumbling speech and dirty ways did not inspire respect. Reports of his blatantly homosexual attachments and his alcoholic excesses were diligently spread back to a horrified countryside... In casual conversation about the Duke of Buckingham, Sir Walter Raleigh could remark that royal favourites were 'frequently commanded to uncomely, and sometimes unnatural, employments'; a preacher at the popular May Day sermon at St Paul's Cross openly referred to the King and 'his catamites'; correspondents from the Court regaled their friends with detailed accounts of drunken orgies in which men and women were staggering and spewing around the King (p. 89)

There is a lot in this quote that has been discredited or at the very least modified and clarified, so let's focus on the most important elements for your question. Do we have evidence of "blatant homosexual attachments", "catamites", and "drunken orgies"? The short answers, respectively are "probably", "not likely", and "no." The second two items can be dealt with more swiftly, so I will address them first.

First, on the issue of "catamites", a very loaded term. The exact source that this phrase comes from was a letter by John Chamberlain to diplomat Sir Dudley Carleton in June of 1619. I unfortunately don't have access to the manuscript version of the letter, but should anyone have access to it, it's archival location is SP 14/99, No. 75. In the version edited by Norman McClure, Chamberlain writes that Isaac Singleton, a canon of St. Paul's Church, gave a sermon on May 1, 1619 at St. Paul's Cross that angered the government enough for them to arrest him. Singleton was "aggreved" by some recent decree, either from King James or Chancellor Francis Bacon, and he "glanced (they say) somwhat scandalously at James and his Catamites as he called them" (McClure, ed. Letters of John Chamberlain, 2:243). There are two important elements about this report. First, that Chamberlain is reporting gossip, as evidenced by his "they say". Second, according to Mary Morrissey in her Politics and Paul's Cross, the claims against the government is directed at Chancellor Bacon (p. 84). The problem with all this is that Bacon was never accused of homosexuality, there is no reference in Bacon's writings and works about this scandal of Singleton, and there is no further mention of Singleton in the state papers. Further, two of the earlier printed editions of this letter, do not contain reference to Catamites, and instead are talking about Bacon's "Latinities" (Calendar of State Papers, Domestic, p. 51; Birch, Court and Times of James I, 2:172). Morrissey makes the mistake of printing both versions in her book, thus clouding the meaning. Without seeing the original, I can't be definitive on what was actually intended, but even if this is a case where the more modern edition is correct (as it probably is) and the text is against James and his "catamites", the use of the term is likely more for rhetorical flair than an actual description of James' relationship with pre-adolescent boys.

There were reports that James was known to be fond of "young" boys, but this typically did not refer to children. It refers instead to men like George Villiers (later Duke of Buckingham; for consistency's sake I'll refer to him as Buckingham throughout), a younger son of a minor gentleman, who was 29 at the time of the 1619 sermon. Many of the later concerns over James affection for young children was created by James' pseudo-adoption of Buckingham's children as his "grandchildren" (Cogswell & Bellany, Murder of King James, pp. 8-22). James affection for attractive young men, typically meant men in their teens and twenties. His physical relationship with them was considered too familiar for the French diplomats that visited the court, but this seems to me to be more about his willingness to touch his servants than in a sexual relationship between king and child (Young, King James & the History of Homosexuality, pp. 80-81). I would argue that Buckingham's relationship with Charles (which was certainly close, though never sexual) at the court of Spain supports this idea. The Spanish were shocked at the level of familiarity between Charles and Buckingham, that Buckingham would call Charles nicknames, or touch him physically. James, Charles, and Buckingham violated rules of etiquette, and it may well be this same lack of decorum that tended to shock foreign diplomats to James' court.

The reference to "drunken orgies" is hyperbolic at best. The main incident Stone is referring to is the reception for King Christian IV of Denmark in 1606. English wit and writer John Harington's letter on the meeting between James and his brother-in-law Christian is often cited as evidence of the excesses of James's court. Harington wrote "we had women, and indeed wine too, of such plenty, as woud have astonishd each sober beholder. Our feasts were magnificent...the ladies abandon their sobriety and are seen to roll about in intoxication (McClure, Harington, pp. 118-121)." Although the revelry over the course of several days was certainly an excercise in feasting and excess, it was not the typical event for James. Much more often, James would be with a small group of men, retreated to a hunting lodge with plans to hawk or hunt the following day. James was a heavy drinker, but never to the excess of "drunken orgies" that offended Professor Stone.

As far as blatant homosexual tendencies, there are three men who are typically associated with James alleged sexual experiences with men. The most thorough study of James' sexual behavior is Michael Young's King James and the History of Homosexuality. Young is committed to the idea that there is enough evidence in the writings of James and his 'lovers' to prove that James had sexual relationships with men, and I am convinced that he is correct on at least one account. It is worth noting, however, that there is no definitive evidence on what constituted James's sexual relationship with men. Put another way, Young does try to decide what physical acts James and his favorites engaged in, whether mutual masturbation or anal penetrative intercourse, but his and any discussion cannot realistically have a definitive answer.

Edit - I made a mistake in claiming that Bacon was never accused of homosexuality. In fact, there is some evidence that Bacon may have been sexually attracted and active with men. The case is no where near as strong as it is for James, lacking in extensive contemporary reports, and letters from Bacon to potential lovers. I do not believe that this fact negates the argument that Singleton's point is unclear and pulled from gossip.