Where does "Whatcha" & "Didja" come from?

Whatcha (or wotcha) is a contraction of a variety of expressions: what are/do/did/have you - see the Wiktionary entry.

Similarly didja.

They arose because that's how people speak.

The original is "What (did/have/are/can, etc) you"

This is pronounced /hwɒt/ (verb) /juː/

The verb is unemphasised and then lost

The /juː/ is reduced to /jə/[or /ja/] the result is

/hwɒt/ /tʃə/

The two [t]s run together,

Watcha - /hwɒtʃa/

The same applies to "didja" where the [d] and [t] [j] combine. Edited 20201015

Whatcha and didja are the 'informal' spellings of what you and did you respectively. People often spell them the way they pronounce these words.
'Palatalisation' is what caused those pronunciations.

Palatalisation is a type of secondary articulation whereby a sound is produced with the front part of the tongue coming near the hard palate.
It's often regressive and is influenced by the following /j/.
In consonants, it mostly affects the alveolar obstruents (/t d s z/) when they're immediately followed by /j/, resulting in palatalised sounds. In IPA, the palatalised sounds are usually transcribed by adding a superscript [^j] after the primary symbol.

Most secondary articulations are allophonic in English i.e. they do not change the meaning of the word. For example, 'tune' is usually pronounced [t͡ʃuːn] in most varieties of British English while most Americans pronounce it [tuːn], both of them mean the same thing.
In some languages, such as Russian, secondary articulations change the meaning of the word, thereby making phonemic contrasts. For example, [s^jok] means 'he lashed' while [sok] means 'juice'. _{[Example from Understanding Phonetics by Patricia Ashby]}

In English,

• /t/ and /j/ coalesce into /t͡ʃ/
• /d/ and /j/ coalesce into /d͡ʒ/
• /s/ and /j/ coalesce into /ʃ/
• /z/ and /j/ coalesce into /ʒ/

Most of time, the clusters /tr/ and /dr/ also undergo palatalisation and become /t͡ʃr/ and /d͡ʒr/, respectively.
So 'true' and 'drive' might sound as if they were 'chru' and 'jrive'.

Examples:

• What you
• Did you
• Tune (in BrE)
• Gradual
• Nature etc.
• Fusion

Words like situation, gradual, perpetual, Tuesday, education etc., have also undergone palatalisation.

Geoff Lindsey in his book English after RP explains that this coalescence/palatalisation had already happened to some words by the time of RP, e.g. culture, future, picture, and soldier, so this is not a random idiosyncrasy.

_{/j/ represents the 'y' sound as in yes.
/ʃ/ is the 'sh' as in ship.
/ʒ/ as in genre.
/t͡ʃ/ as in church.
/d͡ʒ/ as in judge.}

Whatcha is as the other answers say. Incidentally, wotcha is either a contraction of the greeting 'what cheer?' (York Mysteries, c.1440) or 'what do ye?' (Chaucer c.1386).

Didja is, as you say, from 'did you?'

Like 'ain't', 'innit', 'wassup', 'bloomin' and scores of others, they're still in use in London and South East England and probably in many other parts of the country. Many such expressions are labelled 'Cockney'.

This feature of connected speech is often called 'yod coalescence'.

Other answers have focused on the sound change, but not on the early usages of these forms in written writing. Whatcha and Didja, as uses in print, emerge somewhere in the late 19th or early 20th century as representations of oral dialogue.

This n-gram shows the origin point within Google's English corpus to be in the 1910s, a time when short stories and serials in popular magazines were flourishing:

Examples from Google Books:

Harper's Weekly, 1 July 1911:

"Whatcha want?" she said. Ferguson gazed dully at the receiver for a full minute, in sodden and abject silence. "Hay, four-o-nine, whatcha want?" the voice went on, somewhat querulously.

Yale Sheffield Monthly, October 1916

"Didja ever see such nerve on any woman?" said Tillie indignantly. "Coming in here, and me showin' 'er all the stock for nothing. Say, Mae, didja go on that party last night?"

Using the newspaper archive America's Historical Newspapers (via Readex), I can find uses from a couple of decades earlier. For example:

Daily Inter Ocean (Chicago, Illinois), December 6, 1896:

"Say, didja come here t' pass remarks 'r t' eat?" demanded the waiter. "Come, whatcha want--er git out."

St. Albans Daily Messenger, 8 June 1877:

"High bluffs, eh? Why, they ain't nothin' to what I see the other night. Why I see Jim Orndorff plank up $250 on king high and get away with the pot. Whatcha think o'that for a bluff?"

As you can see, these uses of didja and whatcha accompanied a style of writing that represented more accurately how the words and phrases were being pronounced.