Wordorigins.org

Hooch, the liquor; hootchy-kootchy, the sexually suggestive dance; and hooch, a hut or dwelling all look similar, but in fact are all from different roots.

The name for the liquor is native American in origin. It is from the Tlingit, after the Hoochinoo Indians of Alaska and a distilled liquor manufactured by them. From Seal and Salmon Fisheries and General Resources of Alaska, 1869:

The natives manufacture by distillation from molasses a vile, poisonous life and soul destroying decoction called “hoochenoo.”

The clipped form appears in 1897 in M.H.E. Hayne’s Pioneers of the Klondyke:

The manufacture of “hooch,” which is undertaken by the saloon-keepers themselves, is weirdly horrible.

Hootchy-kootchy, a suggestive dance, somewhat short of a striptease and often performed as a carnival or side-show attraction is of unknown origin.

The word first appears as a nickname for a minstrel entertainer in 1890. What relation this has with the later uses is not known. From Biff Hall’s 1890 The Turnover Club:

I have been told that one night “Hoochy-Coochy” Rice, the minstrel man—they always call Billy “Hoochy-Coochy” because he invariably says that whenever he comes on stage—entered Hoyt’s room...and stole a new song.

The sense meaning a suggestive dance appears in 1895 in James Thornton’s song The Streets of Cairo or The Poor Little Country Maid, about a young woman going unescorted to the World’s Columbian Exposition in Chicago in 1893. The “streets of Cairo” is a reference to one of the exhibits at the fair that featured belly dancers, among other attractions:

She never saw the streets of Cairo,
On the Midway she had never strayed,
She never saw the kutchy, kutchy,
Poor little country maid.

The familiar form of the word appears in the 1896 Amherst College Olio:

Ide and Ward entered the one mile hoochee coochee.

Finally, the sense meaning a hut or dwelling is more recent and is from military slang. Originally, the term was hoochie and is probably a borrowing from the Japanese uchi, house. The term comes courtesy of the American post-war occupation forces and those stationed in Japan during the Korean War. It first appears in 1952, in the San Francisco Examiner of 26 October:

The “hoochie” is a GI term for a bunker or a prepared defensive position.

Also from 1952 is Oscar Brand’s ribald song Lee’s Hoochie on the album Out of the Blue about an airman who contracts a venereal disease from a prostitute, Miss Lee:

Way down in Seoul City,
I met a Miss Lee.
She said, "For a short time,
You can sleep with me."

I went to her hoochie,
A room with hot floor.
We left our shoes outside,
And slid shut the door.

Within a decade, the clipped form hooch had appeared. From Richard Tregaskis’s 1962 Vietnam Diary:

A lot of hooches (native huts or houses).

(Sources: Historical Dictionary of American Slang; Oxford English Dictionary, 2nd Edition)