It is cerainly not uncommon to see Black lining listed in inventories and accounts (e.g. those of Sir John Howard). Also you may have differing qualities of cloth for the lining of the arms as to the body.If you are unsure stick with natural or bleached linen, according to the quality of the doublet.
As to the price of 'black' cloth ... it can be BOTH expensive and cheap.
Sounds odd ? ... maybe but you can get a cheap black using the tannin in Oak Galls (or even Oak chippings) when mordanted with iron (it's why Oak chests get black lines around the iron work. However as this is tannic acid it will cause the fibres of the cloth to rot and it isn't the most 'fast' (i.e. permanent) dye as it will fairly rapidly degrade to a greeny-brown. So these cheap dyes would be used for cheap cloths ... incl. some Linens and Fustians.
To get a good 'true' black you need to use alot of dyes and overdying ... which is both expensive in materials and in labour. Also repeated overdying will require a cloth capable of withstanding the punishment this involves ... so again the cost goes up as you would need a quality cloth for the process and why waste the money/effort on the dye if the cloth doesn't make the investment worthwhile.
This last point holds true for most cloths/colours. Cheap dyes are used for cheap cloths, expensive dyes (eg grain) are reserved for high quality cloths.
As to 'Jacks' ... there are actually quite a few references which suggest that the outer layer of some cloth armours was black (e.g. the double de fense in the Howards accounts is covered with Black Fustian) but I've only ever seen pictures where they are white.
Cheers
Dave