From there it probably gets a lot more divergent. I'd also bet that cells and something like DNA is pretty common wherever carbon-based life is found, but my understanding is that the exact way DNA codes for proteins is pretty arbitrary, and the fact that all life on Earth more-or-less uses the same coding scheme is an artifact of the common origin of life rather than because it matters.
Body plans and the details of multi-cellular life are probably going to be wildly divergent, with some caveats. For instance, eyeballs evolved multiple times on Earth, so it would be surprising if eyeballs didn't evolve pretty often elsewhere.
(Incidentally, if aliens are made of proteins, sugars/starches and fats, that means that whatever they look like, we can probably eat 'em, barring the usual toxins and allergens.)
So, you ask, what about Silicon? Silicon also has 4 valence electrons, so it ought to be a nice substitute, right?
The problem is that Silicon is in the next row down.. so it's more massive and it's electron cloud takes up more room. So you still only have 4- connectors, but it's a larger element.
There's clearly an advantage into being able to form precise and small molecules. Carbon, with it's small size and 4 valence electrons, is a clear choice for a backbone.
I'm a complete layman, but how does that play out in planets with gravity force significantly stronger or weaker than Earth? Woul that skew which elements make the best building blocks?
Silicon is basically impossible as a building block for life. It forms bonds that are too strong, meaning it requires much more energy to fuel life processes and reducing the rate of chance collisions which lowers the prospects for abiogenesis.
Carbon is the only realistic choice; it is the only element abundant enough with the right balance of stability and versatility.