If it really is more like a field, then your plan to seed the lawn may only be as effective as repainting a damp wall. In other words, a complete renovation may be needed, because if the grass isn't growing very well, there must be a reason.
For better root penetration, the area for the grass should be deep and loose. Only grass with aerated roots will grow well. You need to eliminate any areas of compacted soil (which can be identified by looking for areas where puddles form, or where there is an increased presence of broad-leaved plantain and grass weed, for example). You can do this by loosening the soil down to a level of at least one spade deep.
Before aerating the lawn, a herbicide should be used to fight the grass weeds. Otherwise, lawn aeration will only have limited success against the weeds. You may also need to repeat the process.
It makes more sense to fertilise the grass after re-seeding rather than before, because this will make it easier for the seed to take root. Wait for heavy rain or water the lawn thoroughly so that the fertiliser you have already applied is better distributed before re-seeding.
Make sure that you use a seed mix that matches the existing lawn, or you could end up mixing ornamental lawn with general-purpose and sports pitch grass. Due to its different composition, ornamental lawn is always finer and denser than general-purpose and sports pitch grass. My tip: Don't be fooled by promises like "for shaded lawns" when making your purchase — there is no such thing as "shaded lawn" seed. Grass is always very much in need of light.
Once the over-seeded grass has taken root and is growing well, continue to fertilise the entire area until autumn, following the fertiliser manufacturer's instructions. Use an autumn/winter lawn fertiliser up until autumn. I recommend treating the lawn with lime next spring at the latest. As you are by the Baltic Sea, I assume that you have sandy soil. The quantity of agricultural lime you need to use should be determined using a fertiliser recommendation based on a soil analysis, which your local garden centre will usually be able to provide (ask in the plant protection department). As a rule of thumb, 50 g of lime per square metre for regular annual maintenance liming; 100 to 200 g (in two or three distributions one after the other) if the ground has not been limed for some time.