If u will
Prince uses this phrase at the beginning of each verse as a kind of concession to the listener.
"If you will" is an idiom that provides a way of conceding to another's presumably different point of view. In "Dig if u will a picture" it means, "try to imagine what I'm imagining." This kind of engagement is a popular technique in pop music. It serves to capture the listener's attention, here, romantically.
Animals strike curious poses
They feel the heat
The heat between me and u
These lines allude to the idea that sex is one of people's more animalistic and base instincts.
For much of Prince's career, one of his biggest selling points with the female audience was his bawdiness. If you remember anything from the music video for "When Doves Cry"—which Prince directed—it's probably the image of a naked Prince crawling out of a bathtub across a petal-laden room filled with soft light and incense.
In any case, this line is interesting because not only is Prince making the connection between sex and primal instincts, he is suggesting that the passion between him and "u" is even more animalistic than animals are used to.
Maybe I'm just like my father
Prince's father John L. Nelson was also a musician—and guess what his stage name was.
Prince's father was known on stage as Prince Rogers…and he later gave that name to his son when he was born. (Prince's nom de reality is Prince Rogers Nelson.)
In the movie Purple Rain, in which"When Doves Cry" appears, Prince's character "The Kid" has an abusive father. As Prince falls in love with Apollonia, he too becomes abusive, tearing his relationship apart.
In Prince's real life there's nothing to suggest that John L. Nelson was an abusive father, though he did leave the family when Prince was ten. He later reconciled with Prince and even co-wrote some of his songs, such as "Computer Blue," which is also on Purple Rain.
Maybe you're just like my mother
Prince's mother Matti Shaw, like Apollonia in Purple Rain, was a singer.
Matti Shaw was a successful jazz singer who went on to produce more than a dozen albums. Like Prince, she debuted at age 19 at the First Avenue nightclub in Minneapolis.
This is what it sounds like
When doves cry
In Christian mythology, the dove is often a symbol of the Holy Ghost, but it also frequently represents the soul in general.
Psalm 123:7 reads, "Our soul is escaped as a bird from the snare of the hunters, the snare is broken and we are delivered." The "snare" is likened to the human body, and the bird is often depicted as a dove.
The pure white dove may be a suitable symbol because of the cultural association of whiteness with purity and innocence. Prince's lyric suggests that he and the woman he is addressing are connected as souls, not just as bodies, making their love potentially incredibly pure and true.
Don't make me chase u
This line departs from the general melodrama of the song in that it is actually quite playful.
It's quiet, but you can make out a chromatic line on a xylophone under this line; in the barebones instrumentation of "When Doves Cry" this is pretty interesting.
Prince wants to make sure that there's emotional breadth to the song—and in a way this playful line serves to justify the sadness of the chorus.
Alone in a world that's so cold (a world that's so cold)
"When Doves Cry" makes use of African-American gospel music techniques such as layering vocals into a choir and "shouting"—adding lyrical lines in response to the main phrase.
Black gospel music began to take on its current sound in the 1930s with Thomas A. Dorsey, who popularized freer forms of Christian choir music through the use of blues progressions and jazz instruments.
"Shouting"—also called ad libbing—became a popular technique due to these influences and has become a sort of calling card in gospel music. Indeed, these shouts are often the most emotional parts of gospel songs.
The same holds true for "When Doves Cry"—while the emotional timbre of the main vocals remains steady throughout the song, the increasing use of shouting builds until the end with Prince pleading to his darling not to cry.