Currently have 46 Construction. What's the cheapest way of getting 70+? How about quickest? Ideas of what I should to to improve my house?

the only reasonable way of doing it is oak larders. It takes around 9k to get to 70.

the cheapest way would be to construct and remove the armour stands in the skill room

 

 

 

if you can get the armour/have the smithing lvl

I'm with mchainmail. Oak Larders are great for XP. They're what I trained on for ages to reach my level (Though I haven't trained Construction since a month after its release :P). But I leveled quickly when training using Oak Larders.