Well, the only example I can think of is:
http://patrikthevampire.com/comic/page-65
He's romanian (I think?? I'm not quite sure about that, he might also be from either Serbia or Bosnia) so he speaks a little bit different but it doesn't feel offensive, nor does is become unintelligible. Instead it adds to the character, making him feel a little bit more fleshed out.
Eastern Europeans doesn't have "w", and have a hard time pronouncing the "th" in "That", "They"... You name it. That's because it doesn't exist outside of the english language (as far as I know) and it's often pronounced differently depending on the word.
So, "That/they" would become "dat/dey" and "thanks" would probably be "Tanks" or "Fanks" depending on the person saying it.
(I don't see it as offensive, mainly because... Europeans really can't pronounce "th" and so that's just how things are?? Tho, I can only speak of myself here when I was abroad as a kid xD)
One thing about bilingual characters is that they (very) often forget certain names of things or translate idioms directly so that they become really... Odd.
Also, depending on their nationality, try to look up videos where they speak English and try to listen in on which words they seem to struggle pronouncing? Maybe that could help a little?