First of all, regardless of whether you are contractually obliged to do so (and you almost certainly will be) you really must discuss your involvement with your family's firm with your employer. If they are of any size or scale they will most likely have a policy concerning outside interests with which you should familiarise yourself.
I would suggest you find a friendly Partner or equivalent and sound them out. You will most likely find that most people have spare time projects be they family businesses, friends with start-ups or charities.
I think the response you get will depend very much on the firm you work for, the nature of your family's business and the proposed level of involvement.
In terms of tax ramifications (and this is my personal opinion rather than advice) so long as your pay from your family firm is subject to PAYE it should be straight forward. If you are or become director you will have to submit a tax return. Simply having two sources of income does not in itself create complications from a tax perspective.
In terms of how to pitch it - follow the guidance provided by the friendly Partner - this will very much depend on your firm's sensibilities and sensitivities.
In my case I am a Director in my family's firm which I cleared with my firm following an informal sounding. This was relatively straight forward; they simply wanted to confirmation that it would not impinge on my work nor that the business was in any way related to consulting or the industry area I work in. Moreover I think they realise that people with outside interests tend to be more rounded individuals and therefore better employees.