Bipolar Transistor Inverter Circuit Analysis
You are so very close to answering your question. You want to know how to calculate the collector current and aren't sure how to develop that value. But let's look at the schematic again:







simulate this circuit &ndash; Schematic created using CircuitLab

Note that I've now included some information that was included in the question that you may have missed that could have helped you. Note that the voltage across \$R_\text{C}\$ is specified!! So you already know that there is \$2.5\:\text{V}\$ across \$R_\text{C}\$, right? From that alone, you can compute the current that must be present in \$R_\text{C}\$. And all of the current in \$R_\text{C}\$ must also then flow into the collector of \$Q_1\$, right? So that's also your collector current.

So you can compute the collector current! And from that, you can now work out everything else from the equations you already wrote about and understand.

I think you are just seconds away from solving your own question.