One bioinformatics method that can help you is the ADME-Tox computation. What you can measure is pharmacological properties test based on Lipinski’s rule of five, bioactivity, drug likeness and drug score, whereas preclinical trials prediction included ADMET (absorption, distribution, metabolism, excretion and toxicity) test, health- effect probability and maximum recommended daily dose (MRDD) prediction. You can check the full ADME-Tox computation protocol in our publication here: Article In silico identification of 2-oxo-1,3-thiazolidine derivativ...
The best way to find out is to measure it directly. This is possible if you are using eukaryotic cells, but very difficult with bacteria. It's quite easy if the compound is fluorescent, allowing you to use flow cytometry, but most are not. Therefore, you have to use sensitive mass spectrometry to measure the amount of the drug in a batch of cells. Some of the drug may be stuck to the outside of the cells, but it is possible to correct for this. You can also use these methods to measure the rate at which the drug enters the cells.
Less directly, you can measure the effect of the drug on the cells. If the target is intracellular and you see the effect of the drug, you know that the drug has entered the cells, although you don't know how much.
If your drug can be labeled, flow cytometry will be a good choice as Dr. Adam highlighted. You may try surface (none- permeabilized cells) or intracellular signal (permeabilized cells). So you may have an idea about the relative proportions that your drug is internalized or on the surface.