In Englishes which allow the usage of ain't, why do we put "no" afterwards?
I ain't got no money.
Your dad ain't got no nose.
Is it compulsory to use no in order to make a proper structure with ain't?
Ain't and "double negatives" (using no, none, nowhere etc with not) are both features of many non-standard varieties of English, all over the world.
Neither of them is used in any standard English; in the non-standard Englishes where they are used, they are often used together, but they can be used separately, as Michael Harvey says in a comment.
The habitual use of ain't no longer occurs in standard Englishes and its use in other varieties is stigmatised by some [misguided souls] as being indicative of lower social status and a lack of education. The history of ain't is interesting as the proscription against it appears, much like the proscription against the perceived phenomenon of so-called h-dropping, to have been the product of a middle class fad.
What may this have to do with the Original Poster's question one may wonder? Well, as can be seen from the first paragraph of the excerpt from Syntactic Variation in English further below, the occurrence of invariant ain't and the occurrence of negative concord (also known as "double negation") are both features of all non-standard Englishes. In other words, in every non-standard variety of English both invariant ain't and negative concord are grammatical. So on this basis, the use of ain't automatically licences the use of further negative items within the clause, representing as it does, basic clausal negation. Notice, however, that this does not mean that the use of negative concord is compulsory in sentences with ain't—which, of course, it isn't in many varieties of English.
It's interesting to consider potential reasons why these two features are ubiquitous in non-standard Englishes. With negative concord, one could surmise that negative concord is prevalent cross-linguistically. Languages which don't exhibit negative concord are the exception, not the norm. However, this does not explain why negative concord is a feature of all Englishes apart from standard English. With invariant ain't, it could be argued that ain't is a product of both phonological processes and the general developmental arc in English of moving towards dropping inflection. However, again this does not explain why ain't should not exist in standard English but exist in all other varieties. Here, however, there is a simple and cogent explanation: it is only in standard English that prescriptivists managed to artificially pervert the natural language in an attempt to socially distinguish its speakers from those considered to be of inferior social status.
Below is the relevant excerpt from Bend Kortmann's 'Syntactic Variation in English: A Global Perspective' (in The Handbook of English Linguistics, Bas Aarts and April McMahon, eds. John Wiley & Sons. 2008. pp. 603—624)
In the domain of negation the two negation features most widely known to occur in all non-standard varieties are multiple negation (or negative concord) and invariant ain’t. But there are other negators, notably invariant don’t and especially preverbal never, which are almost equally frequent. The other two negation features presented below [N5—7] are used in considerably fewer nonstandard varieties:
Multiple negation / negative concord (e.g. He won’t do no harm, I couldn’t say nothing about them, I’ve never been to market to buy no heifers). The frequency with which multiple negation is used in individual nonstandard varieties may vary greatly. In white dialects of American English, for example, frequencies have been found to vary between 50 and 80 per cent (Schneider 2000: 219). A striking pattern Anderwald (2002: 109–14 and unpublished work) has found in corpus-based studies of England, Scotland, and Wales is a south–north cline, with rough proportions of multiple negation usage of 40 to 45 per cent in the South of England, 30 per cent in the Midlands, and around 10 per cent in the North of England, Scotland and Wales. Interesting variation can also be found for syntactic and lexical constraints on multiple negation in different varieties (e.g. in African American Vernacular English multiple negation crosses clause boundaries, indefinite constituents of embedded clauses being marked negatively because the predicate of the superordinate clause is marked negatively; Schneider 2000: 219).
Ain’t. Invariant ain’t in present tense declaratives, questions, and tags represents a neutralization in the negative between be (e.g. I ain’t going out tomorrow, They’re all in there ain’t they?) and (auxiliary) have (e.g. I ain’t had a look at them yet, Gotta be lucky at something, ain’t you love?), as well as a neutralization of person distinctions of Standard English. In some varieties, especially pidgins and creoles, there is a tendency to extend the use of ain’t to full verb have (e.g. Ain’t you trouble with your car?). In fact, in African American Vernacular English ain’t is also used as a full verb negator equivalent to don’t/doesn’t and, especially, didn’t (e.g. sumpin’ I ain’t know about, You ain’t expect to find her over here, did you?; Schneider 2000: 214–15). In some pidgins and creoles, ain’t (or: in/en/eh) has even acquired the function of a general (i.e. tense-independent) preverbal negator (e.g. Trinidadian English The girl eh lie ‘The girl didn’t lie’) as further described in (N5).
Here is a relevant Wikipedia article on ain't, in which the section Proscription and stigma gives a brief overview of the social history of ain't.