I am designing a solar system with the first planet located very close to the parent star.
Probably start with the basic "ballpark" expression for a planetary surface temperature ( the Effective Temperature ) :
$$T_{eff}=\left[ \frac {A_{abs}}{A_{rad}} \frac {L(1-\alpha)}{4\pi\sigma\epsilon D^2} \right]^{\frac 1 4}$$
For a tidally locked body $\frac {A_{abs}}{A_{rad}} = \frac 1 2$
It's a little easier to work with quantities related to Earth and the Sun :
$$\frac {T_{planet}}{T_{earth}} = \left[ 2 \frac {L_{star}}{L_{sun}} \frac {1-\alpha_{planet}}{1-\alpha_{earth}} \frac 1 {R^2_{planet}}\right] ^ {\frac 1 4}$$
where the distance from the star ($R_{planet}$) is in AU and the $\alpha$ values are albedo values and the luminosity values ($L$) can be in any unit as long as it's the same for both (it's common for these values to be quoted relative to the Sun in astronomy).
The factor of two comes from Earth having not being tidally locked compared with your planet which is.
This planet has become tidally locked to the star.
Is it possible for the sun facing side to have a molten lava ocean ?
In short yes, the central area will certainly be able to do this, although the boundaries for the end of this ocean would depend on a lot of details (it's practically a science paper in itself to calculate for any given body).
What would the conditions be like for on the other side of the planet, where the surface is still solid ?
Solid but what temperature ? This is tricky, but there's a more significant problem.
Remember you have to have an atmosphere (and would have a ghastly one) for such a body to exist with any possibility of having life.
The closest thing in real life to such a body would be an exoplanet called Trappist-1f although it's not quite what you're looking for.
The "hotspot" would be ballpark 1400 K and the opposite side's "cold spot" would be of the order of 219 K (i.e. extremely cold). Somewhere along a narrow band there might be a region where it's livable.
However the atmosphere is a problem. It's going to generate extremely strong winds (on that exoplanet and on yours). It has to - you've searing heat on one side and freezing cold on another. The resulting atmospheric circulation (which has to happen) would be violent.
It would also transport gases from the lava ocean (none of them friendly to human-like life) to the otherwise habitable regions. I can't see a way to avoid this. Given you're looking for an ocean sized hotspot of lava, this would be really hellish even in the "temperate" regions.
Could a settlement be built on the side facing away from the sun ?
Maybe, but probably a lot, lot easier (and safer) to build an larger orbiting settlement. Any settlement (for human like species) would be essentially sealed and would need excellent climate control internally to cope with fluctuations in climate.
There's also a potential issue with seismic activity. You've a planet heated to molten temperatures for the crust at one side and cold as the worst of Earth's polar conditions on the opposite side. This is bound to generate seismic (tectonic) activity on a significant scale. I'd imagine the "temperate" region to be worst affected as this is the region where the cold plates meet the warm plates.