He really has two choices:
1) Do something that is likely legal and constitutional, that will seem to do more than it actually does, such as:
a) Opening the NICS system to private persons doing private sales on a voluntary basis. There are several ways in which this can be done, and as long as it is not mandatory, I see nothing wrong with it as a policy. Can Obama legally do this on his own? Maybe.
b) He could direct his DOJ to consider anyone selling more than "X" number of firearms to be "engaged in the business of selling firearms", thus requiring them to have an FFL. This would have to be tested in criminal court. It likely could go either way - but Obama would be out of office before a legal precedent could be established.
2) Obama could do something much more radical and unconstitutional, such as:
a) Requiring a NICS check on all sales
b) Banning sales of some firearms
My gut feeling is that he will go with option one - something that will not be blatantly unconstitutional, but will allow him to say that he has taken action to greatly expand background checks. However, we shall all see very soon.