I don't understand what you mean by being convicted for 5 years. If you were fined, this offence will not be spent for 5 years, assuming you are not convicted of other offences.
Without knowing what the food offence was, it is unlikely that this will count against you in obtaining permission to sell alcohol. You can apply for a Premises Licence and have tons of convictions, the Licensing Authority cannot take them into account. However, to get a Personal Licence, which you need to be a Designated Premises Sopervisor, relevant convictions (listed in the Licensing Act 2003) MAY prevent you. The only food related convictions which are relevant are:-
An offence under either of the following provisions of the Food Safety Act 1990 in circumstances where the food in question is or includes alcohol�
(a) section 14 (selling food or drink not of the nature, substance or quality demanded);
(b) section 15 (falsely describing or presenting food or drink).
These will only be relevant to your application if the Police believe is is a problem. You do NOT need a Personal Licence to run a premises, you can employ a PL holder to supervise for you.
I think you have no problem but further information would be useful.