Hi
You definitely need to clean the old thermal paste off i5 and heat sink
Clean i5 before removing it from the socket.
Do not apply too much paste, a pea size is too large
I always apply paste to heat sink if the area of contact on heat sink is smaller than surface of CPU
.
There is debate whether to apply a thin uniform layer or a few rice grains in the centre and let pressure distribute the thermal material
Intel seems to favour the second for the type they put on standard heat sinks
ArCtic silver 5 has a good reputation
http://www.tomshardware.co.uk/thermal-paste-performance-benchmark,review-32804.html
Another Toms hardware review said salad cream was quite good !!
Regards
Mike Barnes