My first thought was to just feed the signal into transistor base (capable of 4GHz) through a 1K resistor.
Um, you should definitely impedance-match your amplifier. That might mean you essentially have something like a close-to-50 Ω terminated line from which you "steal" your base current through some larger base resistor, but it should definitely not simply be "signal line -> 1 kΩ -> base".
and add low/high-pass filters later.
Since a transistor used for power amplification will not operate within a linear range: Not later, now. A non-linear device is a mixer, and you're producing every product (and squares) of the harmonic signals within your input, which means components at DC, at twice your carrier frequency, three times... and if you got multiple tones inside your signal, the difference and sum and difference-of-sums frequencies of these.
In the end, neither us nor you nor the regulatory bodies involved want that you get beef with the ENS.
My assumptiom is that alufoil wrapped around the wire to my breadboard,
RF and breadboard are already a bad combination. RF, breadboard and amplification are a recipe for unexpected emissions and crosstalk. Don't use breadboard. Get a piece of copper-clad PCB material, and use a knife to "cut free" traces on that, and solder your components onto that.
If you can (that's already a bit tricky), use a trace width that implies good impedance matching (assuming you have a second copper plane on the bottom side of your board).
There's also testboards for RF transistors which you can order, or download the manufacturing files from the producers of these transistors. Also, it's not that hard to design a small board, e.g. with eagle or kicad, that has the right trace width, a ground plane, easy-to-solder holes or contact pads and contacts for edge coaxial connectors, and then order these for < 5$ from places like OSHPark.
