Logan Killicks is Janie’s first husband. Yup, that's about it.
We're only partially joking—very little of Logan’s internal feelings are given, and he may strike you as a pretty one-dimensional character. But, this lack of characterizing information is appropriate given the emptiness and despair he symbolizes to Janie...and what little we know of Logan is less than pleasant.
He's old and ugly—in sharp contrast to Janie’s youthful beauty. Janie resents him for his asymmetrical face and for lacking the etiquette to wash his feet before coming to bed (fair enough):
"His belly is too big too, now, and his toe-nails look lak mule foots. And ‘tain’t nothin’ in de way of him washin’ his feet every evenin’ before he comes tuh bed. ‘Tain’t nothin’ tuh hinder him ‘cause Ah places de water for him. Ah’d ruther be shot wid tacks tan tuh turn over in de bed and stir up de air whilst he is in dere. He don’t even never mention nothin’ pretty."
She began to cry.
"Ah wants things sweet wid mah marriage lak when you sit under a pear tree and think. Ah…" (3.26-28)
If Janie represents beauty, Logan represents everything ugly there is—in terms of all the senses.
In fact, Janie sees him as a "desecration" of her vision of true love, based on her experience underneath the blossoming pear tree. He represents the antithesis of her notion of love, and Janie goes into the marriage with skepticism veiled only by her naïve, young hope.
Logan is also emotionally destitute. What little affection he shows Janie at the beginning of the marriage is described as "speaking in rhymes"—he's basically buttering her up with sweet nothings. After that, he only shows anger and frustration when Janie resists his attempts to command her:
Six months back he had told her, "If Ah kin haul de wood heah
and chop it fuh yuh, look lak you oughta be able tuh tote it inside. Mah
fust wife never bothered me ‘bout choppin’ no wood nohow. She’d grab
dat ax and sling chips lak uh man. You done been spoilt rotten."
So Janie had told him, "Ah’m just as stiff as you is stout. If you can stand not to chop and tote wood Ah reckon you can stand not to git no dinner. ‘Scuse mah freezolity, Mist’ Killicks, but Ah don’t mean to chop de first chip." (4.1-2)
Logan seems to have the idea that marriage means dominating a woman and that women are objects for men to put to use. He also feels unappreciated by Janie because she doesn’t worship him for making her the mistress of 60 acres of land. (Boo hoo, right?)
However, Logan isn't an evil character—he's just a sullen guy with feet that smell like blue cheese. His ineptitude with words doesn't do justice to the thoughts swirling around in his grizzled head. When Janie threatens to leave him, Logan feels genuine fear...but because such fear is incited by a woman, he doesn't have the vocabulary with which to respond.
So, instead of directly addressing the issue and possibly putting himself in a position of vulnerability, Logan instead tries to brush off Janie’s threat and belittle it. As a result, Janie feels that he's insensitive and doesn’t value her, so she leaves Logan to elope with Joe Starks.
Hey, at least Joe Starks washes his nasty feet.