No, you won't be able to do that, because you will still be entering the UK through Ireland (because you landed in Dublin).
The reason you cannot enter through Ireland and then go on to the UK is because there is no immigration control between Ireland and the UK (due to the Common Travel Area). So, if you land in Dublin, you will go through Irish immigration and will be stamped into Ireland.
Then when you go in to the UK (either Northern Ireland or Liverpool), you will not go through UK immigration, and so there will be no record of when you arrived in the UK, because your passport won't be stamped/scanned by UK immigration.
To avoid this happening, what you need to do is book a flight that lands in the UK and NOT in Dublin or Shannon.
That way you will go through UK immigration, not Irish immigration.