The easiest way will be the tube. You will need an Oyster card or contactless card, or you can buy a ticket from one of the machines either at ground level or at platform level. From memory I think the price is around £1.50. It is two stops in the tube, and the tube runs every 8 minutes or so. Here are some directions from the station to the HGI, walk takes around 6-7 minutes.

https://www.flyertalk.com/forum/27265665-post70.html

As an alternative the 423 bus goes to Hatton Cross from T5 too and is free. However, it will take a bit longer and only runs every 30 minutes at that time in the evening.

Getting to T3 I would say getting there around 7am, you can aim to arrive by 7.30am and still be fairly comfortable. Tube is frequent that that time in the morning. Again tickets can be bought from the machine in the station if you don't have an oyster or contactless card. It is only one stop so takes a few minutes.

Also do have a look at uber. It may cost a few pounds more but tat may be outweighed by convenience.

The one thing I am unsure of is whether you will need to collect luggage. Certainly for a T5-T5 connection your bag can be stored overnight in the terminal baggage store, but as you are doing T5-T3 I don't know if that would work - as far as I know T3 baggage system doesn't have facilities to store bags overnight.