Core and winding dimensions for an electromagnet and bifiliar winding
I found this calculator on the web. It seems to use the correct formula and that is: -

Force exerted = \$(NI)^2 \times \mu_0 \times\dfrac{A}{2g^2}\$

Where 


N is number of turns
I is amps
A is cross sectional area of coil
\$\mu_0\$ is permeability = \$4\pi\times 10^{-7}\$
g is the gap between the solenoid and the thing it attracts to


Note that the permeability of the material you wind it on does produce more flux density but the permability of the gap is the totally dominant part of the equation and material permeability is not included because it is so very minor. 

An an electrical analogy, having something with high conductance (ferrite) in series with something with low conductance (air) still means something with low conductivity i.e. air is dominant except when the gap becomes small as in gapped ferrite transformers or inductors (which you haven't got)

Doubling the turns or doubling the current produces 4 times the force but doubling the gap quarters the force. Doubling the diameter of the solenoid produces four times the area and hence 4 times the force too.

Increasing the wire diameter allows more current to flow (for a given power supply voltage) BUT only if the supply can give the extra current. Ideally, wind with silver wire (costly) thus keeping the resistance low and feed as much current in as you can. Make the coil diameter as big as you can too. As many turns as you can manage also.

Bifilar winding won't help - this will be a dc solenoid with maybe a low frequency excitation and skin effect will be almost negligible - better to use two wires in series because N doubles rather than split current into two parallel paths (in effect this is just one turn still).