How to activate the mining process in the standard bitcoin client? I have done the whole synchronization and now it should be mining bitcoins from what I understand but I don't see anything showing that it is actually mining for bitcoins nor any activity from my cpu… Is this normal ?

Do I need to enter a command to start the mining process or is the process done automatically without having to do anything ?
Warning: doing what is described below will not gain you any coins at all with exceedingly high probability, and just burn electricity (see other answer).

Bitcoin Core still has a built-in miner. It is not efficient, not optimized, does not use any special hardware (like special CPU instructions, GPUs, FPGAs, or ASICs), and is only useful for testing on test chains.

You can enable it by issuing the RPC command setgenerate true, and disable it with setgenerate false. These commands can be issued in the debug console in Bitcoin-Qt, or using an RPC client (like bitcoin-cli) to send it to a running bitcoind instance.