A climbing aircraft needs less aerodynamic lift than in horizontal flight, not more.
Now I have your attention, I hope. The reason is quite simple:
Lift equals weight, and just because the pilot choses a different flight path angle, the weight of the aircraft does not change. The total of all lifting forces must still balance the weight, but in climb you get a small lifting contribution from the engine(s) because its (their) thrust will point upwards just like the rest of the airframe.
Don't let the many arrows and greek letters confuse you. To be in equilibrium, lift (L, dark blue), drag (D, red), thrust (T, green) and weight (m⋅g, black) must add up such that they can be combined into a closed run of vectors. I've done this with the lighter-colored vectors around the weight. Since the flight path points upwards, so does the thrust which now has a small vertical component. The lift vector can be a little shorter now.
Consider the extreme case of vertical climb: Now all thrust supports the weight, and aerodynamic lift is no longer needed.
There is a second, much more subtle effect: When you climb, air gets thinner and engine performance goes down proportionally. At the same indicated air speed, the aircraft will continually decrease its climb speed, and this deceleration frees up a tiny inertial force, which again adds to lift and counteracts weight.
Conversely, at the beginning of a climb phase the aircraft needs to create momentarily more lift to accelerate itself upward. Only then, when climb speed increases, lift must be bigger than weight to overcome the inertial effect which at this moment works downwards. For the supernerds: If you integrate the lift deficit over time of the aforementioned effect and the extra lift over time for climb acceleration, both cancel exactly.
To answer your question directly: To climb you need to increase excess energy, not speed. This is normally done by increasing engine power output, or by trimming the airplane at a lower speed where drag is less, so more power remains for climbing. This question contains more details on how to get an aircraft to climb. Note especially @SteveV.'s bucket analogy.
If you use the airplane's kinetic energy as its source of thrust, the same mechanism can be applied to instationary climbs, where speed is traded for altitude, like in gliders.
The nose-up attitude is simply the result of a different flight path. Since the required aerodynamic lift will be almost the same, the angle of attack will also be almost the same and the whole aircraft needs to fly nose-up. This is similar to a car which has the same attitude towards the road, but when you drive uphill, both car and road will be tilted upwards.
This analogy breaks down when you change speed - flying at lower speed needs more angle of attack to still create the same lift, and this nose-up change will be added to your attitude angle.