If wavefunction is just a probability function, how does an electron interfere with itself I have read lots of quantum mechanics books. 
The chapters that are talking about De Broglie, lots of them name the chapter as "Wave-particle duality" and says: "Electrons are both waves and particles". So I start to think that (for example) an electron sometimes becomes wave sometimes becomes particle. 
But when I start to read the chapters about Schrödinger and wavefunction thing. I see that that wave nature thing does not belong to electron itself. It is about its location. So there is no wave particle duality. Electron is always a particle. But the location of electron is represented as wave, because of uncertainty principle. So electron can't interfere with itself because it is always a particle. The interference is about its location function.
1-) Are these all true?
2-) If true, if an electron is always a particle, in double slit experiment how does an electron interfere with itself without observer, but there is no interference pattern with observer?           
 A: Saying that things "are both waves and particles" is a vestige of the 18th century way of thinking, and really ought to be done away with. Everything is described by a wavefunction. Period.
What is a wavefunction? It is a complex-valued function. If you are interested in an electron's position, it is easy to think of it as a complex-valued function of space. It alone is not a probability. Only its square magnitude gives a probability distribution. This process of taking the square magnitude explains how interference occurs. Just like with classical waves, two out-of-(complex-)phase wavefunctions can add destructively, such that taking the square magnitude after adding them results in a value close to $0$.
When people say things are both waves and particles, they mean that in some limits the object acts indistinguishably close to that of a classical particle. But these are just simplifications in certain cases; they have no bearing on the fundamental nature of the objects in question.
A: An electron can interfere with itself. there have been experiment of interference with single electron.
Saying that an electron is alway a particule is then wrong. 
The wave function is the "best" way we have find to describe electrons and other quantons. object at the quantum level are not wave or particles. They just follow quantums rules and waves or particules are just two limits of this description.
A: See this answer and the comments. There is an explanation how occures fringes due to the EM field from the surface of the slit or wire:
https://commons.m.wikimedia.org/wiki/File:Moellenstedt_biprisma_voltage_shadow.JPG
It shows the influence of an electrical field to fringes.
https://commons.m.wikimedia.org/wiki/File:Moellenstedt_biprisma_schematic_arrangement.JPG
This shows how experiment with electrons was arranged in generally.
As a result you may find out that the potential of the material which forms the slits is responsible for changes in fringes dimensions. Ergo can we say that the potential is responsible for the fringes at all?
And there is a relative question too:
Whenever we can observe photons immediate, they are particles. That includes that photons have a inner structure with periodically varying electric and magnetic fields. The EM field of a radio antenna exists because this field consists a lot of photons with their periodically changing EM components. Whenever we observe statistical manifestation of interaction between photons and certain physical states (most of them based on diffraction) we interpret the fringes on a screen as waves manifestation of particles. And at the same moment we always emphasize that these states are not observable. It's an interpretation of what we see.
To interpret the fringes as a result of the interaction between photons (or electrons, ...) and the EM field of certain physical states is not common but has some charme. No more need in interference of an electron (or photon) with itself in the single particle experiments. No more sentences like "we can mathematically write down but not describe what happens".
Yes we always describe that photons interfer EM fields (static between condenser plates or in interaction with other particles or especially with other photons (photon bunching)) but we don't articulate that. We repeat what physicists understood 90 years ago. We came to QED and work with the quantization of fields but our verbal expression is from 1920.
Are there new concepts for the explanation of the wave-particle duality?
A: 
So there is no wave particle duality. 

By particle we tend to use the classical macroscopic definition of a particle, which means a shape( a volume) a center of mass that describes absolutely the particle's position in (x,y,z,t). A billiard ball, for example.

Electron is always a particle. But the location of electron is represented as wave, because of uncertainty principle. 

The electron is an elementary particle , a basic building block of macroscopic matter as we know it.
A single electron in an observation, as in this single electron double slit experiment,


Electron buildup over time

is a dot in the picture. Thus each electron as it passes the slits ends up as one dot on the screen, of dimensions about a micron, having a specific ( x,y,z) , thus similar to a classical particle.
BUT as the electrons accumulate we see an interference pattern similar to the classical wave mechanics. The distribution though is of a population of electrons, not the same electron, and it gives a probability distribution that can be used to give the probability of finding the next electron at a specific (x,y).
Probability distributions have the same meaning both in the classical and quantum mechanical  case. One cannot nail down a prediction, in a similar way that there exists a probability curve for life expectancy, given one's age, but it is not a prediction of death, except if one is 103   where the probability goes to .999999 ( a distant cousin of mine died at 103 the other day). 

So electron can't interfere with itself because it is always a particle.

The electron is a quantum mechanical entity, described by a wave function and quantum numbers.

The interference is about its location function.

For a single electron given an ( x,y,z,t) there exists a probability it will be found if an experiment is carried out at that point in space time. Probability distributions can be seen experimentally by repeating the experiment with different electrons, not the same one.

1-) Are these all true?

In the qualified way described above.

2-) If true, if an electron is always a particle, in double slit experiment how does an electron interfere with itself without observer, but there is no interference pattern with observer? 

You are talking of checking which slit the electrons go through as they pass. When one puts detectors at the slit , it is a different experiment, different wave functions.
The wavefunction of an electron going through two slits is one type of problem, a solution of the quantum mechanical equation with electron+two slits as boundary conditions. Putting detectors at the slits changes the boundary conditions. If grossly enough the interference is lost because the wavefunction is different. There have been experiments with minimal detection at the slit which clarify this: the greater the effect of detection the smaller the interference pattern.

We have realized a which-way experiment closely resembling the original Feynman’s proposal exploiting focused ion beammilling to prepare two nanoslits and electron beam induced deposition to grow, selectively over one of them, electron transparent layers of low atomic number amorphous material to realize a which-way detector for high energy electrons. By carrying out the experiment in an electron microscope equipped with an energy filter, we show that the inelastic scattering of electron transmitted through amorphous layers of different thicknesses provides the control of the dissipative interaction process responsible for the localization phenomena which cancels out the interference effects.

A: It is just a misleading concept that electron is particle or a wave ... the answer is neither particle nor wave but BOTH. How we came to know that electron is particle?? Suerly through experimental evidences. A man is affected by its very imediate surroundings .. and what is in surroundings .... ?? Most oftenly matter.    Electron is particle or a wave is just our theoretical imagination that how electron might look like ... but our picture for an electron being a particle as well as wave is consistent with experiments so we say that our theory works and fit all assumptions what we made for electron .. so important is measurement out come. 
