Since this is a German employment contract, you will have to pay both social security contributions and income tax to Germany, just like any other employee in Germany.

I can do the payroll accounting, i.e. calculate every month how much in social security deductions and income tax is due and file the monthly social security announcement at the German institutions.

Your non-German employer (or I for them) would have to apply for a Betriebsnummer (company registration number) at the Bundesagentur für Arbeit (labour bureau), under which the social security contributions would be paid.
Getting a Betriebsnummer in itself does not mean that your company establishes a permanent establishment in the tax sense in Germany, it is just something that is necessary to correctly allocate the social security contributions.
Your non-German employer would only establish a permanent establishment in the tax sense (which would mean owing German tax on the part of their profits generated in Germany!) if they fulfill the conditions set down for that in the double taxation agreement between their country and Germany. Most of these conditions are logical, e.g. if your non-German employer has an office in Germany through which they carry out business, that would be a permanent establishment and therefore taxable in Germany. However, there is also a more insidious condition that also leads to a permanent establishment: it’s enough if you as their employee “have, and habitually exercise, an authority to conclude contracts on behalf of your employer” and they will suddenly have a permanent establishment in Germany.
So if your employer wants to avoid taxing a part of their profits in Germany, they had better not give you such authority.

But let’s leave the higher spheres of international tax law and go back to the nitty-gritty of German payroll accounting.

Since your employer is non-German, they are not obliged to deduct the German income tax from your salary every month and forward it to the Finanzamt, like German employers have to, §38 (1) EStG.
So it will always fall to you to pay the income tax that’s due on your salary to the Finanzamt.

Regarding German social security contributions, the law does not differentiate between German and non-German employers.
So your non-German employer would – in theory – have to deduct the employee part of the German social security contributions from your salary every month, and forward them (and their own employer part of the social security contributions) to the proper German social security institutions.
However, in practice, nearly all non-German employers use article 21 no. 2 of the EC regulation 987/2009 which allows them to come to an agreement with you, the employee, to forward the social security contributions for them. And you will agree to do so if you want that job.
It is a good idea to already include this agreement as a clause in your German employment contract, should the German social security institutions ever ask to see it.

In summary, there are two options:

• very unlikely: that your employer transfers the social security contributions (their employer part and your employee part) directly every month to the German institutions (or if your employer is from a SEPA country, gives these institutions permission to take that money out of their bank account).
  Your employer then transfers what’s left of your gross salary, i.e. (gross_salary – employee_social_security_contributions) to you.
  The German income tax is paid by you to the Finanzamt after your yearly income tax return for the first year. Ideally, you will have been saving how much I told you to save each month, knowing that that money wasn’t really yours and that you will have to forward that tax money to the Finanzamt in the end.
  From the second year, you prepay your German income tax every quarter to the Finanzamt, on 10. March, 10. June, 10. September and 10. December.
  I strongly suggest that you give the Finanzamt a SEPA-Lastschriftmandat, i.e. permission to take the income tax out of your bank account themselves, that way you can never miss a payment deadline.
• nearly always: your employer transfers you your gross salary, i.e. without any deductions, plus the employer’s part of the social security contributions, and you transfer the entire employer and employee social security contributions (AG=Arbeitgeber=employer and AN= Arbeitnehmer=employee) every month to the German institutions (or give these institutions permission to take that money out of your bank account, which I strongly suggest).
  The German income tax is paid by you to the Finanzamt after your yearly income tax return for the first year. Ideally, you will have been saving how much I told you to save each month, knowing that that money wasn’t really yours and that you will have to forward that tax money to the Finanzamt in the end.
  From the second year, you prepay your German income tax every quarter to the Finanzamt, on 10. March, 10. June, 10. September and 10. December.
  I strongly suggest that you also give the Finanzamt a SEPA-Lastschriftmandat, i.e. permission to take the income tax out of your bank account themselves, that way you can never miss a payment deadline.