Can Einstein's constant explain expansion?

I read somewhere that Einstein or Newton believed that the universe was completely static, where it neither expanded nor contracted, but simply remained fixed. It was concluded that due to attraction force between any and every particle in the universe, the universe would ultimately collapse in on itself. In order to compensate for this, a repulsion force for gravity was theorized (the proverbial anti-gravity), which would as the distance between particle increased the force would also increase. The theory was later rebuked, one reason being that if one particle slightly moved closer or further away from the other, the repulsion or attraction forces would overpower the other and the universe would either expand or contract, thus contradicting the static universe theory.

However, as we now know the universe is indeed expanding at an accelerated rate, can it be that this is due to the repulsion forces of gravity overpowering its attraction forces?

Just as a footnote question: Is the acceleration of expansion linear or increasing?

The simplified model of the universe used in discussions like this is the FLRW metric. From the tone of your question I'm guessing that you're not looking for a detailed mathemtical treatment, so let me just pull out the equation that answers your question. This is one of the two Friedmann equations:

$$\frac{\ddot{a}}{a} = -A\rho + B\Lambda \tag{1}$$

(this is a simplified form that ignores pressure, which is a good approximation for all except the very early evolution of the universe).

The left side of the equation is basically the acceleration of the universe. So if the left side is less than zero the expansion rate is decreasing and if the left side is greater than zero the expansion rate is increasing. If the left side is exactly zero then the expansion rate is constant.

On the right side of equation (1) $A$ and $B$ are positive constants, and $\rho$ is the density of matter while $\Lambda$ is the cosmological constant.

Note that on the right hand side the term $-A\rho$ and $B\Lambda$ have different signs, so they oppose each other. If we make the cosmological constant $\Lambda = 0$ then we get:

$$\frac{\ddot{a}}{a} = -A\rho$$

So the acceleration is negative i.e. the expansion is slowing down. This makes sense, because matter attracts matter due to gravity, so if there's only matter in the universe we expect its mutual gravitational attraction to slow down the expansion. Now what happens if we take out all the matter so now $\rho = 0$? We get:

$$\frac{\ddot{a}}{a} = B\Lambda$$

This time the acceleration is positive i.e. the expansion is speeding up. The cosmological constant in effect creates a gravitational repulsion that pushes the universe apart. This is the repulsion force that you talk about in your question.

Einstein's original model adjusted the constants $A$ and $B$ so the matter density and cosmological constant terms exactly balanced and the acceleration was zero. With some other adjustments this leads to a static universe that is neither expanding nor contracting. However, as you say, this turns out to be an unstable equilibrium and in any case it's a rather artificial construction and one that I'm sure Einstein was glad to see the back of.

However, as we now know the universe is indeed expanding at an accelerated rate, can it be that this is due to the repulsion forces of gravity overpowering its attraction forces?

and the answer is that yes, since around 6 billion years ago the $B\Lambda$ term has been bigger than the $A\rho$ term, so for the last 6 billion years the expansion rate has been increasing. Your last point is slightly more complicated:

Just as a footnote question: Is the acceleration of expansion linear or increasing?

In the above I've assumed that the repulsive term is a cosmological constant. If this is true then the rate of acceleration will eventually become constant. At this point the universe will be expanding exponentially i.e. it will double in size every $x$ years, where $x$ is a constant.

However it's possible that the repulsive term is more complicated than a cosmological constant. For example it could be a form of dark energy called quintessence. In that case it is not necessarily constant, and the acceleration of the expansion rate may increase or decrease. We have no firm theories of what quintessence is or how it behaves, so it's impossible to comment at present.

The experimental evidence is currently compatible with a simple cosmological constant, though these are hard experiments and the precisison isn't great enough to rule out quintessence.

The accelerated expansion of the Universe is an effect of repulsing forces BY DEFINITION. The nickname of the reason of these forces is "dark energy".

The problem is that Einstein's $\Lambda$ term is very simple addition to equations. It could be added even if Einstein didn't want to make Universe static. So it is a great possibility, that Einstein didn't guessed.

Note, that until acceleration was discovered, $\Lambda$ term was regarded as completely wrong. And it was a surprise, when it appeared useful. Nevertheless it is very simple addition without any weighty grounds.

So $\Lambda$ term models a global intrinsic property of our Universe.

There are other possibilities, like quintessence. The difference is that quintessence can travel from place to place causing repulsion being bigger or smaller in different regions of space.