I've always thought it was a parodic reference to the prologue of The Canterbury Tales:
Whan that aprill with his shoures soote
The droghte of march hath perced to the roote,
And bathed every veyne in swich licour
Of which vertu engendred is the flour;
Whan zephirus eek with his sweete breeth
Inspired hath in every holt and heeth
Tendre croppes, and the yonge sonne
Hath in the ram his halve cours yronne,
And smale foweles maken melodye,
That slepen al the nyght with open ye
(so priketh hem nature in hir corages);
Thanne longen folk to goon on pilgrimages, ...
which in modern English looks something like this:
When April with his showers sweet with fruit
The drought of March has pierced unto the root
And bathed each vein with liquor that has power
To generate therein and sire the flower;
When Zephyr also has, with his sweet breath,
Quickened again, in every holt and heath,
The tender shoots and buds, and the young sun
Into the Ram one half his course has run,
And many little birds make melody
That sleep through all the night with open eye
(So Nature pricks them on to ramp and rage)-
Then do folk long to go on pilgrimage, ...
(from http://pages.towson.edu/duncan/chaucer/duallang1.htm )
so Eliot's providing a contrasting view of Spring / rebirth / sex:
APRIL is the cruellest month, breeding
Lilacs out of the dead land, mixing
Memory and desire, stirring
Dull roots with spring rain.
Winter kept us warm, covering
Earth in forgetful snow, feeding
A little life with dried tubers.
e.g. contrast
'April ... stirring dull roots with spring rain'
with
'April with his sweet showers' piercing March 'to the root' (balls-deep) and 'bathing' her with his 'liquor'.