How to find density of element metals
One way you can do this is by looking at the packing structure of the metal.

As an example, if you look at Wikipedia, you see that Tungsten has a body-centered cubic crystal structure. This means that in each unit cell, there are going to be two Tungsten atoms. We can then predict the density of a perfect Tungsten crystal lattice using some geometry and unit conversion.

First off, I'll give you an equation which you can prove to yourself quite easily so I won't go into that. Density of a crystal is: $$\rho=\frac{n*M}{N_A*V}$$

Where, $n$ is the number of atoms in the unit cell, $M$ is the molar mass of the atom, $N_A$ is Avogadro's number, $V$ is the volume of the unit cell.

So, for Tungsten this comes out to be $$\rho=\frac{2*183.83 g*mol^{-1}}{6.022*10^{23}*(\frac{4*139*10^{-10}cm}{\sqrt{3}})^3}=18.45\frac{g}{cm^3}$$

The experimental density of Tungsten is $19.33 \frac{g}{cm^3}$.

The answer is usually a little better than that, but still quite close.

The only information you need to do this calculation which isn't on a periodic table is the packing structure and the atomic radius.

Something which is noteworthy is the atomic-packing-factor, $APF$, which comes from finding the ratio of volume of the atoms to volume of the unit cell and represents how much space the atoms fill in the cube, or how efficient the structure is at packing.

For the body-centered cubic (BCC), $$APF=\frac{2*\frac{4}{3}\pi r^3}{(\frac{4r}{\sqrt{3}})^3}=0.68$$

That means BCC, takes op 68% of the total available space per unit cell for equal sized spheres.

Check out this link if you want more info on that.

So, to answer the actual question which how do we find a trend with all this, we now know that density depends on radius, which we already have a trend for, molar mass, which also has a very simple trend, and packing structure, which is the real unknown.

There is this from this page,


  In the resonating valence bond theory, the factors that determine the choice of one from among alternative crystal structures of a metal or intermetallic compound revolve around the energy of resonance of bonds among interatomic positions. It is clear that some modes of resonance would make larger contributions (be more mechanically stable than others), and that in particular a simple ratio of number of bonds to number of positions would be exceptional. The resulting principle is that a special stability is associated with the simplest ratios or "bond numbers": 1/2, 1/3, 2/3, 1/4, 3/4, etc. The choice of structure and the value of the axial ratio (which determines the relative bond lengths) are thus a result of the effort of an atom to use its valency in the formation of stable bonds with simple fractional bond numbers.
  which I don't actually understand but seems to explain why certain lattices are chosen.


Basically, using the fact that radius decreases going right and molecular weight increases going right, we would predict that density would increase uniformly across the periodic table for elemental metals, except that various metals pack in different ways. Hexagonal Close Packed is the most efficient packing system, so I wouldn't be surprised to find that being associated with many high density metals.

I hope that gives a good idea of how there is sort of a trend, but also why no trend is truly there.

EDIT:

To figure out which has the highest density, I would start by figuring out which pack in a Hexagonal Close-Packed structure as that is the most efficient packing structure with an $APF$=.74