Turkish seems to use two different language suffixes: -ce and -çe although they seem to have got mostly formed into new words in their own right. Can anyone explain why this is and how it has come about?

For example:
Türkçe - The Turkish language
İngilizce - The English language

How about this: ç and k are unvoiced, while c and z are voiced. I guess there's consonant assimilation process here.

This is according consonant harmony. This is consonant harmony:

"When two consonants are next to each other, if the first consonant is a voiced consonant then the the one next to it is also voiced, if the first consonant is unvoiced then the the one next to it is unvoiced as well."

Maybe we can say the same thing in a better way:

"Voiced consonants match with their same kind when two consonants come next to each other."

I think my second definition sounds better.

There are eight voiced consonants: p,ç,t,k,h,s,ş,f

In fact just because the consonant harmony rule all suffixes that start with a consonant have a voiced version. For instance -di > -ti (past tense), -de > -te (locative case) etc.

http://www.turkishclass.com/grammar_consonantHarmony.htm

There you go