As you say, an electron has negative charge. Photons have no charge. Since the charge of a collection of objects is just the sum of their charges, an electron cannot consist only of photons.
At our current state of knowledge, an electron is an elementary particle, which cannot be said to consist of anything else. maybe someday in the context of string theory (or some other deeper theory) people will be able to talk about what an electron consists of, but not yet.
The weird thing about electrons is that the space around them is full of photons and "virtual" e+e- pairs. These pairs pop into existence very briefly and exit, but the positrons are attracted by the electron's field and the electrons are repelled. This screens the electric charge of an electron a bit by polarizing the surrounding vacuum. In fact, if you scatter two electrons off of each other, how close they get to each other depends on the energy of the collision. It has been observed that the strength of the electromagnetic interaction increases with increasing energy for exactly this reason, and so this odd kind of "dressing" of an electron is experimentally verified. If you take the argument to the extreme, saying that an electron appears to have a larger charge the harder you hit it, it in fact has a formally infinite charge at infinite energy. But fortunately, we know how to do the calculations by using only the observed, finite electric charge.
It could well be that there is some small radius or some high energy at which an electron no longer seems to be a point particle. String theory promises to remove the infinities of a theory by proposing particles like electrons are very tiny strings. So far there is no way to test this, but we're on the lookout for good ideas.
A lot of this weird stuff was worked out in the 1930's and 1940's -- the vacuum polarization argument was first correctly done by Hans Bethe, who sadly passed away just a few weeks ago.
(republished on 07/13/06)