Although The Professor will give some tips on getting upgraded, upgrade opportunites, especially on trans-Atlantic flights, are few and far between. For instance, I have been an American Airlines customer since 1984, have flown over 1,600,000 miles with them, and have NEVER received a free trans-Atlantic upgrade.
But there are ways to get upgrades. Let's look:
Use Miles: Airlines will permit "free" upgrades, based upon availability, if you have enough miles to do so. Airlines only make a few seats in first class available on overseas flights, so the key is to request them very early (flight schedules and seating come out 330 days before the flight) or 30 days before departure.
Be Elite: Enroll in an airline elite status program like American Airlines AAdvantage. In short, American's program works like this: the more you fly, the more upgrade "coupons" you earn. You can redeem those coupons for free upgrades on domestic flights only. Here are the details: https://www.aa.com/i18n/aadvantage-program/miles/redeem/award-travel/500-mile-upgrades.jsp. When using my "coupons" to upgrade, I try to select flights that don't have a lot of business travelers who have way more seniority in elite programs that I do.
Negotiate: Cancelled flight? Schedule change? Overbooked flight? These might give you some negotiating tools to ask for a better seat. It always help if you have elite status with an airline.
Begging, Being Nice to Airline Crew/Agents, Medical Excuses: they don't work.
Have some tips? I would love to hear them.