Is the universe expanding at a speed of almost $2c$? I've been told nothing can travel faster than the speed of light. Therefore, from my vantage point the diameter of the universe is increasing at a rate of $2c$. Are there any flaws in my thinking?
 A: 
Are there any flaws in my thinking?

Several, but the most important one is this:  the spatial metric is not an object with a world line.  Thus, one cannot conclude that the metric expansion of space is 'speed' limited (whatever that may mean).
Also, I suspect that you're picturing the expanding universe as an expanding ball with a diameter and a boundary but, if so, that would be a flaw in your thinking too.
A: There is no specific speed with which the universe is expanding.
Instead the rate of expansion increases with distance between the points being considered, in accordance with Hubble's Law.
At great enough distance, the rate is greater than c, greater than 2c, greater than any given value.
There is no radius for the entire universe, although the observable universe could be considered to have a radius.
The radius of the observable universe does not match-up with expansion at the speed of light, because the rate of expansion varies with time.  Expansion has decelerated in the past and is thought to be accelerating currently.  A photon emitted by an object receding at greater than c may eventually reach a region of spacetime that is receding at less than c, permitting to make progress toward us and be observed.
We can currently observe light from stars that were and still are receding from us at greater than c.
See the following papers for further discussion:
http://users.etown.edu/s/stuckeym/AJP1992a.pdf
https://arxiv.org/abs/astro-ph/0011070
https://arxiv.org/abs/astro-ph/0310808/
A: 
"I've been told nothing can travel faster than the speed of light."

What you've been told is an ambiguous statement. What exactly is meant by "travel"? A more accurate statement is that nothing can overtake a photon. That also holds true for expanding space.  
A: You need to be careful about exactly how is the speed limit defined. You cannot travel faster than a light ray in a local reference frame. But the geometry of spacetime changes over vast distances in the universe, so the relative "speed" of such distant objects (like galxies on the opposite sides of the observable universe) cannot be compared in a simple way. The geometry of spacetime is expanding and it "drags" the galaxies with it. But nothing travels faster than light in a small region where the expansion is negligible.
Also please note that the edge of the observable universe (which is called the particle horizon) is not receding at the speed of light, but at 3 times the speed of light.
You can find a discussion of this subject for example here.
A: 
I've been told nothing can travel faster than the speed of light.

That's correct; but of course only as far as it's correctly understood.
A bit more explicitly and carefully there are two separate correct statements to be considered:

*

*(1) in the context of geometry in general and (the geometric-kinematic aspects of) the theory of relativity in particular, the exchange of "light" (between some particular pair of participants; "sender" and "receiver") is understood as the signal front of any signal exchanged between them; i.e. whichever way the receiver first learns something about the sender (the receiver changing its state in response to the sender having changed its state; where both sender and receiver are typically thought of as systems constituted of electro-magnetic charges, whereby any such "light" signal having been exchanged can generally be considered an electro-magnetic signal). And


*(2) as far as sender and receiver were and remained at rest to each other, such that they could determine and agree upon being characterized (as a system) by some particular value of distance between each other (rather than, for instance, merely being rigid to each other and being characterized by a pair of quasi-distance values), such that they could determine values of (average) speed "between" each other (as system), whether it might be the (average) speed of a signal front they exchanged, or the (average) speed of some particular thing (or identifiable traveler) which had been passing between them, or the (average) speed of some common occurance (or phase) which they both observed without thereby necessarily exchanging a signal between each other,
then they determine the speed of anything whose passing by implied that sender and receiver thereby exchanged a signal necessarily as less than, or at most equal to, the speed of the corresponding signal front (which is usually denoted as "$c_0$"; or often just "$c$".).

Therefore, from my vantage point the diameter of the universe is increasing at a rate of "$2~c$".

That's a flawed statement since it doesn't seem to use the symbol "$c$" as speed strictly in the sense of (2), i.e. in reference to some particular sender and some particular receiver (e.g. "you") having been and remaining at rest to each other.
(This kind of flaw seems common among people who refer to "distance" when trying to characterize subjects of cosmology; not explicitly heeding Synge's advice that
"For us time [duration] is the only basic measure. Length [distance] is strictly a derived concept".

[J. L. Synge, "Relativity. The general theory", p. 108]
.)
A: The main thing here is that expansion of universe is not the same thing as the speed which is limited by $c$. The expansion itself has no limit in principle (see other answers for more information).
A related fact is that even without cosmic expansion, it is possible to find examples of distances which change at $2c$.
If two things approach you from opposite directions, each at speed $c$ relative to you, then the rate of change of the distance between them, as observed by yourself, is $2c$. This does not break the speed limit because nothing is moving at that speed (relative to you or anything else).
For example of the above, if you stand somewhere near the LHZ particle collider at CERN, then a bunch of protons approaches you from the right at close to $c$, and a bunch of protons approaches you from the left at close to $c$. You could (in principle) stretch a tape measure between these two bunches of protons. The distance between them, as indicated by your tape measure, is then changing at the rate $2c$. But note that nothing is moving at that speed relative to you. And in the rest from of each proton beam, the other one has speed very close to $c$, not $2c$. Confused? Well all I am doing here is showing a few relevant facts; for explanations you will have to keep learning.
