I am in an unhappy relationship and have one young child with this man.  We are only together because of our son, but everything else is dead between us. 

Well if your relationship has been useless for a long period then you should be looking for someone else. 

I have found myself becoming more and more attracted to my boss, who is in a relationship and has children. 

If he is in a relationship and has children then he is less likely to give things up and move on.  It would have to be someone very tempting for him to give up his current relationship.  One thing you need to find out though is whether his relationship is also on the rocks.  If it is then that gives you an increased chance that he might want a relationship with you.  If it isn't then it might mean that he would want you as his mistress and without telling his current partner (of course even a mistress type relationship can eventually lead to a long term relationship, but not always).

I personally don't recommend breaking up someones family but if a guy was going to end up breaking up because he was very unhappy in his relationship then maybe you should go for it. 

We only speak about work related issues when we meet but I can't help thinking there is a mutual attraction.  He is very very professional, and would never ask staff anything personal about themselves,

Bosses have to be very careful what they do and who they do it with.  Alarm bells start ringing in their head about sexual harrassment and for that reason he will be immensely cautious about any relationship with any member of staff.
 

but I find when we do chat we have a good laugh and he has started to prolong eye contact with me, which makes me flustered and blush.

Prolonged eye contact alone is not conclusive evidence that he fancies you, friends and close colleagues can also prolong eye contact.  And as he is getting to know you more and more, the close eye contact is likely to increase.

What do you advise?

By what you have said I don't think either of you know each other enough yet to make any judgements or any significant moves.  He will certainly be cautious about making any moves because of his position as boss.  However there are a few things you can do to indicate to him that you don't mind him around.  Like allow him to move close to you, you move close to him, even some accidental touching, of course you should only do this when you are comfortable.  Also work some more personal things into the conversation.  But overall you need to continue to progress without pushing things too fast.  Don't just walk in, grab hold of his leg and say "fancy a date" because he won't be ready for any of that yet, you need to work his mind into thinking of you as relationship potential or sexual potential, whichever you prefer.