To upgrade your Dynasty members require funds from your Swiss bank account. You probably won't get this until late in the campaign. Make sure you issue the Building Permit edict in every mission and try and stick the reward from each completed objective in to you slush fund. When you build banks, have them siphon money from the treasury to you, etc. Basically exploit every avenue for increasing your personal wealth.
You can only upgrade your Dynasty members once per mission for a reasonable price, e.g. you'll be able to upgrade for $20,000 but if you want to upgrade again while playing the same mission the price will be $150,000 or more depending on the level you're upgrading to.
To get this quickly, start up a sandbox game in the modern era with a construction based winning condition. Go in to your dynasty with LT+Y, upgrade your chosen dynasty member then go back to the mission and complete the winning condition. Boom, mission completed. Repeat another three times and you'll have your level 5 dynasty member.