Recently, I am also looking into this approach.
It seems that carbon nanotube (CNT) fibers might be a promising option in the near future of 'rocket-launch-via-cable', due to its excellent conductivity and high strength.
--------- Here are some ideas and calculations ----------
Consider the electric cable made of Carbon Nanotube Fibers with diameter $1mm$ and length $600km$, targeting a space launch to International Space Station (orbit $408 km$, velocity $7.7066 km/s$).
The weight of the cable will be $π (0.5 mm)^2 (600 km) (2100 kg/m^3) = 989.6 kg$, where $2100 kg/m^3$ is the density of Graphite.
Winding the cable and put it on the rocket, with one end connecting to the ground of electricity supply and drop (release) the cable gradually from the rocket during launching so that the relative speed of the cable is zero to the ground during releasing.
Now, the calculation for the necessary power.
Suppose the total weight of the rocket is $10,000kg$, with orbit speed $7.7066 km/s$ and acceleration of $3 g$, the power needed to accelerate the rocket during launching is $10,000kg \times 4 g \times 7.7066 km/s = 3.018 GW$.
This is necessarily the maximum power as when the rocket reaches high altitude, its weight will decrease (gradually) and acceleration is much smaller than 3g.
Let's see whether the carbon nanotube can afford this power. Consider using Alternating Electricity with voltage $1000kV$, which is already possible in reality. https://en.wikipedia.org/wiki/Electric_power_transmission#Advantage_of_high-voltage_power_transmission
We need the cable to hold electric current of $3.018 GW / 1000kV = 3018A$.
The ampacity of a single CNT is $10^9 A/cm^2$, which corresponds to maximum $2,500,000A$ for our cable (1mm diameter), sufficient enough. While in practice, the tested CNT fiber only gives $10^5 A/cm^2$, but with CNT-Copper composite, it can be $10^7 A/cm^2$.
(Data from "High Ampacity Carbon Nanotube Materials" https://pdfs.semanticscholar.org/4831/a85d7d32e170ab0eb3639da68aa5ed2de03c.pdf)
Moreover, the cable diameter can be designed so that the closer to the ground, the larger diameter. Then at around $30km$ altitude, we could insert a device (drop from rocket) on the wire to transform the voltage to $10MV$ or even bigger for the remaining cable. High voltage for high altitude could be an option in terms of air breakdown field ($3 MV/m$).
Here are some possible problem.
i) The heat on the wire might be a problem.
The conductive resistance of nanotube is 1.0×10−8 Ωm, which is 0.04 Ωm in our cable (0.5 mm diameter). Then the heat on 1 meter wire in one second is (3018A)^2*0.04Ω = 364 KJ, without considering the 'capacitive reactance'.
My calculation might not be accurate.
ii) The cable in the air might be broken (possibly by wind).
As the cable only has diameter 1mm, it does not seem to be strong, even a strong wind might break it. The released cable in the air will drop over time (might not be the main problem due to air resistance and thin cable), which implies we have to drop more cable than the real altitude. $600km$ of cable length for orbit $400km$ launch is just an estimation, while it depends on a lot factors of how long the cable should be. Also, we need an extra device to retrieve the cable after launching, maybe we can use some battery-powered motor to collect the cable.
iii) Dropping speed of cable from rocket (relative to rocket) is too high, $7.7066 km/s$.
Maybe some tricks of winding the cable in a smarter way will reduce the dropping speed and flywheel tech could be applied.
iv) Electric motor to power the rocket.
Finally, the biggest challenge I think is how to build a light-weight powerful Motor to power the rocket with the electricity from the cable, especially in high altitude where air density is low. This might be solved by considering a two stage rocket launching while the cable approach is only intended for the first stage, and apply hydrogen fuel for second stage.
Another way might be compress the air during launching, and using electricity to accelerate the compressed air to high speed in high altitude. Also, the oxygen could be collected from air if two stage rocket is adopted.
Some more discussion might be of interest.
v) The cloud at $10km$ might be a problem.
In humid air, $1000KV$ voltage will (might) break down. An extra device above the cloud to transfer the voltage may be an option. From ground to cloud, apply $100kV$ voltage but with larger current, hence thicker cable.
--- update ----
Ohm heat is corrected, and now it is $364KWm$, which is not possible for 0.5mm diameter cable.
The solution might be raising the voltage from $1000KV$ to $20MV$ and hence decreasing the current from $3018A$ to $150.9A$, hence the new heat will be $910Wm$.
vi) It is a challenge to transfer energy with only one single wire.
From Tesla's invention, the principle behind is that the rocket could act as a capacitor, while it is not possible when the rocket reaches high altitude. Also, the capacitor of the cable in the air may affect that principle.