You can, but it will almost certainly mean giving up your German citizenship. The Germans are very weird about dual nationality -- I think they see dual nationals as a potentially dangerous fifth column. My mum has never applied for UK citizenship for precisely this reason (my dad is English). She still has her German passport and there's no real disadvantage apart from not being able to vote in UK general elections, and occasional trips to Belgrave Square to get her passsport updated/renewed.
There are circumstances in which you can have dual nationality despite Germans frequently and confidently telling me it is not possible, I have German and British passports (legally!). Your children would almost certainly be entitled to dual nationality, though might have to opt for one nationality or the other at 18 I think the legislation has changed since I was a child. The Germans really don't like it though -- I've had some very strange reactions on the odd occasion when officialdom has realised my circumstances. Sample quote: "Ja, wie denken Sie sich das denn, dass sie die Vorteile der deutschen Staatsbürgerschaft geniessen und sich ihre Pflichten entziehen wollen?" (this was in the course of a discussion about a GCSE certificate, just to put you in the picture...).
Tbh, in these days of EU, there's not really much point in having dual citizenship of two member states, although it has occasionally been useful for me to have two passports (travelling on one while the other was elsewhere having a visa application processed, that sort of thing). I did look at getting German citizenship for the children when dd1 was born, but frankly after half an hour perusing the paperwork required (beglaubigte this that and the next, amtlich anerkannte pi pa po) I'd lost the will to live, shoved it all in a drawer and forgot about it.