Is there anything my dentist can do about the black showing on top of my crown?

It is true that your tooth can turn very dark following a root canal. There are a few reasons that you may have that darkness at the gumline. 
1. The crown has a metal substructure and the very edge of it can show appearing very dark at the gumline. (It is unlikely that this type of crown was used if it was done recently, but worth asking)
SOLUTION: 
Have the crown replaced with a material that is metal-free.
2. The crown is all ceramic containing no metal but stops just shy of where your gumline begins allowing the dark root to be exposed.
SOLUTION:
The margin or edge of where the crown rests can be prepared higher up on the root of the tooth, beneath the gumline
3. The crown is all ceramic, extends past the gumline covering the root but it is made of a ceramic material that allows the dark root to show THROUGH where the ceramic gets thinner and thinner near the gumline. 
SOLUTION:
The tooth can be sanded down or "prepared" a little more aggressively allowing the ceramic to remain thick enough to mask the darkness
You may want to consider having internal bleaching through the same small hole that was used to perform your root canal. Its a very simple procedure that can lighten up the treated tooth making it easier to mask. Clear communication between the doctor and the lab about the need to mask this dark root will also help in solving the problem. It's commonly dealt with successfully, so don't give up.

The only practical way to improve the appearance is to lighten the shade of the tooth. The way to do this is to do internal bleaching, which involves placing whitening solution inside the root of the tooth and sealing it in and letting you walk around over a period of a few days. This will unfortunately involve damaging your crown if you were to request it now, so I normally do this whilst the temporary crown is in place. Attempts to simply make the crown longer to cover more of the root always fail because the higher up the root you go, the darker it tends to get. Some recession (the gum moving away from the crown) is almost inevitable and so this way there will always be a black line - but only higher up the root. Good luck with whatever you decide to do -Kind regardsThomas Dey, BDS, MSc