I'm gonna try for the Kingdom, if I can
"Kingdom" here probably refers to the Kingdom of God in the Christian tradition.
The Kingdom of God, also called the Kingdom of Heaven, is, according to Encyclopædia Britannica, "the fulfillment on Earth of God's will." Various other scholars have defined the Kingdom of God as a kingdom of love, according to the teachings of Jesus regarding God's will.
Perhaps what Lou Reed uses the "kingdom" to signify here is the idea of heaven as a pure ecstasy, more than any more scholarly understanding of "the kingdom." Throughout history, religion and drug use have both commonly been associated with feelings of euphoria and ecstasy, and sometimes, the line between religious experience and drug experience has been blurred.
In the 1960s, people were doing plenty of research—and "research"—into the spiritual effects of recreational drug use. While most attention was devoted to Eastern religions like Buddhism, countless papers, books, and essays have examined the connection between drugs and spirituality. Aldous Huxley's The Doors of Perception described the religious effects of mescaline, an alkaloid found in peyote, a drug which has been used by some Native-American groups for spiritual purposes for thousands of years.
Ph.D. Hutson Smith mentions in his paper, "Do Drugs Have Religious Import?" that the Aztecs described certain "sacred mushrooms" as "God's flesh," and notes further examples of sacramental drug use in Greek, Hindu, Zoroastrian, and aboriginal traditions. If you have access to online journal databases or a really good library, countless other examples of the connection between drugs and religious experience can be found in Philippe de Félice's Poisons Sacrés, Ivresses Divines.
When I put a spike into my vein
A spike is—you guessed it—slang for a syringe.
Part of the lyrical success of "Heroin" derives from Reed's use of street slang. The lyric conveys an almost shocking sense of authenticity, a feeling that Lou Reed was really there, plunging a needle into his vein.
The song sounds like a naked soul speaking plainly of the agonies and ecstasies of life as a junkie in '60s-era New York.
And I feel just like Jesus' son
It's unclear here whether Reed is singing "Jesus, son" or the possessive "Jesus' son."
"Jesus, son" makes less grammatical sense than "Jesus' son," but then "Jesus' son" doesn't make a whole lot of semantic sense, since Jesus didn't, you know, actually have a son.
Either way, Reed is playing with some powerful imagery here. Saying that putting his heroin needle into his vein makes him feel like either Jesus Christ or the mysterious son of Jesus Christ is a pretty arresting—and controversial—statement, and one that could be read in several different ways.
Reed might be saying that his drug use makes him feel divine, or all-powerful, sacrificial, or agonized. Or all of the above.
On a great big clipper ship
Clipper ships were 19th-century sailing ships known for their speed.
A specific type of clipper ship, the China clipper, was the fastest commercial sailing vessel ever built, reaching a top speed of about 16 knots. (That's about the same speed as today's modern oil tankers travel.) The clipper ships were built in America and Britain for transporting passengers and goods over long distances.
So, what are these antique ships doing in this song? Reed probably invokes them to complement the "old-timey" imagery that begins with his wish that he'd been born 1,000 years ago, and the general sense that his journey is taking him to a very different kind of place.
Because a mainer to my vein
"Mainer" is another slang term for the needle.
According to the not-exactly-unimpeachable experts at Urban Dictionary: "Usually this slang is used by people on needle drugs to describe their experience with the needle without saying the word 'Needle.' Their friends and family may use it to try to reach out to them."
I don't really care anymore
About all the jim-jims in this town
What the heck is a jim-jim?
Once again, there are few sources that define the word jim-jim, none of which are entirely respectable. "Jim-jim" would appear to be a term describing a con artist, hipster, or other undesirable person.
Whatever the definition was in 1964 to 1965 when Lou Reed wrote "Heroin," the general sense seems to be similar. The pairing of politicians—in the subsequent line—with "jim-jims" in the sense of con artists would make sense with the common pairing of politicians and crooks in our language and culture.
And, to generalize even more, doubling words to create new words (as in "dodo," "so-so," "dumb-dumb") typically belittles the subject in the English language.
And all the dead bodies piled up in 'Nam
That is, Vietnam. By the time that Lou Reed had written the song (the earliest version dates to July of 1965), protests against the Vietnam War in America had barely begun to pick up steam.
With the Gulf of Tonkin incident and the subsequent escalation of the Vietnam War by Lyndon Johnson in the summer of 1964, protests against the escalation of U.S. involvement became a bit more pronounced, although the full-fledged student uprising against the war was still a few years away.
Perhaps the most notable pre-"Heroin" protest was one held at the University of California at Berkeley, in May 1965. At that time, Cal students held the first draft card burnings, torched an effigy of Lyndon Johnson, and staged a teach-in attended by some 30,000 people.
Fewer than 3,000 American soldiers had been killed by in Vietnam at the time that these early protests took place, though that number would eventually rise to nearly 60,000.