Hi,
I'm a 26 year old German student of English literature who speaks English very well but has a seriously strong accent. I don't think there could be a better person to answer this thread ;P
First of all, I agree with the poster above who said that oftentimes, spelling out phonetic differences can be annoying/gimmicky. I tried with a Russian character once and it ended up farcical, but that was alright in that case since the character was mostly comic relief anyway.
Now, I have two options for you: either I can give you my own impressions on how you could transcribe phonetically accented English spoken by a fluent German native speaker, or I could simply link you to a rather long sound file of myself reading something.
I happen to have a sound file like that lying around on my server anyway. It's me reading one of my older short stories, and it's full of pronunciation mistakes.
I'm not going to link to that, though, without being asked for it, because that would just feel... weird.
So here's my short list of things that Germans do when speaking English:
1) Endlautverhärtung. Isn't that a pretty German word? It means "final consonant devoicing." Without giving you a proper lecture in phonetics (which I'm not sure at all you'd need), it means that certain shifts appear in the end of words in German (and a lot of other languages; I know, for instance, that Polish does the same thing). Some of these shifts are:
The problem with this is that it's partly a lie
(and here is where the lesson in phonetics begins). The "voicedness" of a consonant depends on a lot of things; the amount of vibration in your vocal chords while producing the sound is only factor, and not the most important either. The length of the preceding vowel, for instance (long vowel: voiced, short vowel: devoiced). However, these things happen automatically in the mind of an English native speaker, so if you write that your German professor says "back" when he means "bag" and "hat" when he means "head", they will know what's going on.
2) w. We don't have it. The sound that is formed by bringing your lips close together and then moving them apart again while air passes through them. We really don't have anything like it, and while it's easy enough for a German speaker to imitate the sound perfectly when he concentrates on it, it will often slip his mind. It's the phonetic error I commit most often these days. Transcribe the resulting sound as a v. Note: this doesn't happen to me very often, especially not in word initial position, where it is very easy to produce this sound. However, there are certain constellations that make the w-->v slip especially likely and in which it happens to me almost all of the time if I don't pay attention. The most difficult of those is post t: twenty, for instance. I almost always say tventy. My girlfriend loves it, but it irks me since I actually do want to speak flawless English SOME day ;P
3) v. Your v is our w. Confusing? It is, to us. In English, v means a voiced labiodental fricative, and f is the voiceless labiodental fricative. Very vs ferry. In German, BOTH f and v are voiceless labiodental fricatives and w is the voiced labiodental fricative. (okay, that's a lie, again; German w is typically produced as a voiced labiodental APPROXIMANT, but hey, I wasn't gonna give you a lecture in phonetics ;P). Hence we sometimes forget to voice English v. This doesn't happen to me very often, but something else related to English v does: when there are many words beginning, alternatingly, with w and v, and when I'm paying special attention to the ws, I sometimes switch the two around or even turn a v into an approximant (I've said "wery well" before).
4) th. Now this is the big cliché concerning Germans speaking English, of course, and I'd really hoped that it was just that--a stupid cliché, an exaggeration of our more clueless English speakers--until I had my first English class in high school. Of course we don't have this sound either, and yes it is at first difficult to produce this sound, but once you've figured that out (and unless it happens in constellations like six
ths, which is insane) it should be trivial! Not so for most Germans, it seems. The common replacement is /z/, while /f/ is a lot less common but not entirely rare. Typically one speaker sticks to one way of avoiding the th sound.
There are a lot more things that German speakers typically get wrong--especially in the area of vowels there is a LOT of variation; the two languages hardly share a single vowel sound EXACTLY (that is to say, German vowels tend to be produced slightly differently from the seemingly corresponding English vowel). My professor of Phonetics wrote an EXCELLENT course book called
The Phonetics and Phonology of English Pronunciation, which is aimed specifically at German students and hence highlights all these little differences. I happen to own it (is it ethical for a professor to list his own book as required reading? I don't know, but this book was so wonderful that I almost changed my major to English linguistics), so if you need more examples, I could provide you with some (these four were cited from memory, so if I got anything wrong, don't blame Professor Barry).
Regarding nonphonetic mistakes I must say Amen to the poster who mentioned prepositions. As recently as two days ago I've had to go and ask a native speaker whether I should use "in" or "at" in front of the name of an internment camp. I do have a good stomach feeling most of the time, and I did tend toward at in this case ("At <whatever camp>, <some female character> received no medical treatment for her pregnancy"), but once I stopped and thought about it, I confused myself. Also, obviously, collocations: you don't do mistakes, you make mistakes. Idioms and proverbs should be pretty obvious as well, but figuring out the German equivalent so your character could translate it literally into English may be a little difficult. You can ask me, or you can google for a list of German proverbs/idioms. For instance, instead of saying "he bit the dust", your character could say "he bit the grass", which is what we say in German.
And if you really want to listen to an authentic German speaker making lots of mistakes, ask me and I'll link you to that audio file.